ASKSAGE: Sage Q&A Forum - Individual question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 23 May 2018 23:05:58 -0500Class groups of number fields obtained by adding p-torsion points of elliptic curveshttps://ask.sagemath.org/question/42396/class-groups-of-number-fields-obtained-by-adding-p-torsion-points-of-elliptic-curves/I am a new user. I am very sorry if this question is inappropriate. I would like to test some of my speculations about the arithmetics of number fields obtained by adding p-torsion points of elliptic curves.
More precisely, suppose E is an elliptic curve over $\mathbb{Q}$ of conductor $p$. Let $L=\mathbb{Q}(E[p])$, the field obtained by adding all p-torsion points of $E$. Are there commands in SAGE that allow us to test whether the p-part of $Cl(L)$ is non-trivial?
For cyclotomic fields $\mathbb{Q}(\zeta_p)$, one can use Kummer's criterion to see whether the p-part of $Cl(\mathbb{Q}(\zeta_p))$. However, I do not aware of any analogous criterion for $L=\mathbb{Q}(E[p])$.
Thank you for reading. Tue, 22 May 2018 13:34:31 -0500https://ask.sagemath.org/question/42396/class-groups-of-number-fields-obtained-by-adding-p-torsion-points-of-elliptic-curves/Answer by dan_fulea for <p>I am a new user. I am very sorry if this question is inappropriate. I would like to test some of my speculations about the arithmetics of number fields obtained by adding p-torsion points of elliptic curves. </p>
<p>More precisely, suppose E is an elliptic curve over $\mathbb{Q}$ of conductor $p$. Let $L=\mathbb{Q}(E[p])$, the field obtained by adding all p-torsion points of $E$. Are there commands in SAGE that allow us to test whether the p-part of $Cl(L)$ is non-trivial? </p>
<p>For cyclotomic fields $\mathbb{Q}(\zeta_p)$, one can use Kummer's criterion to see whether the p-part of $Cl(\mathbb{Q}(\zeta_p))$. However, I do not aware of any analogous criterion for $L=\mathbb{Q}(E[p])$. </p>
<p>Thank you for reading. </p>
https://ask.sagemath.org/question/42396/class-groups-of-number-fields-obtained-by-adding-p-torsion-points-of-elliptic-curves/?answer=42404#post-id-42404Please give us a special case. So far, try `EllipticCurve?` to see how ot initialize an elliptic curve in sage. There are many examples. $E[p]$ can be "computed" more or less starting for intsance from the corresponding `division_polynomial`. For instance:
p = 3
for a in [3..22]:
print a
E = EllipticCurve( QQ, [-1, a] )
P = E.division_polynomial(3)
L.<a> = NumberField(P)
Cl_L = L.class_group()
if p.divides( Cl_L.order() ):
print "E =", E
print "RANK ::", E.rank()
print "P =", P
print "L =", L
print "Cl(L) ~", L.class_group()
break
The above rudimentary search finds:
E = Elliptic Curve defined by y^2 = x^3 - x + 19 over Rational Field
RANK :: 1
P = 3*x^4 - 6*x^2 + 228*x - 1
L = Number Field in a with defining polynomial 3*x^4 - 6*x^2 + 228*x - 1
Cl(L) ~ Class group of order 15 with structure C15 of Number Field in a with defining polynomial 3*x^4 - 6*x^2 + 228*x - 1
Wed, 23 May 2018 05:23:29 -0500https://ask.sagemath.org/question/42396/class-groups-of-number-fields-obtained-by-adding-p-torsion-points-of-elliptic-curves/?answer=42404#post-id-42404Comment by tungnt for <p>Please give us a special case. So far, try <code>EllipticCurve?</code> to see how ot initialize an elliptic curve in sage. There are many examples. $E[p]$ can be "computed" more or less starting for intsance from the corresponding <code>division_polynomial</code>. For instance:</p>
<pre><code>p = 3
for a in [3..22]:
print a
E = EllipticCurve( QQ, [-1, a] )
P = E.division_polynomial(3)
L.<a> = NumberField(P)
Cl_L = L.class_group()
if p.divides( Cl_L.order() ):
print "E =", E
print "RANK ::", E.rank()
print "P =", P
print "L =", L
print "Cl(L) ~", L.class_group()
break
</code></pre>
<p>The above rudimentary search finds:</p>
<pre><code>E = Elliptic Curve defined by y^2 = x^3 - x + 19 over Rational Field
RANK :: 1
P = 3*x^4 - 6*x^2 + 228*x - 1
L = Number Field in a with defining polynomial 3*x^4 - 6*x^2 + 228*x - 1
Cl(L) ~ Class group of order 15 with structure C15 of Number Field in a with defining polynomial 3*x^4 - 6*x^2 + 228*x - 1
</code></pre>
https://ask.sagemath.org/question/42396/class-groups-of-number-fields-obtained-by-adding-p-torsion-points-of-elliptic-curves/?comment=42419#post-id-42419My computer has run for about 2 hours without any results. I think one particular reason is that the $Gal(L/\mathbb{Q})$ is always $GL_2(F_p)$-which is already big for p=11 or p=17.Wed, 23 May 2018 23:05:58 -0500https://ask.sagemath.org/question/42396/class-groups-of-number-fields-obtained-by-adding-p-torsion-points-of-elliptic-curves/?comment=42419#post-id-42419Comment by tungnt for <p>Please give us a special case. So far, try <code>EllipticCurve?</code> to see how ot initialize an elliptic curve in sage. There are many examples. $E[p]$ can be "computed" more or less starting for intsance from the corresponding <code>division_polynomial</code>. For instance:</p>
<pre><code>p = 3
for a in [3..22]:
print a
E = EllipticCurve( QQ, [-1, a] )
P = E.division_polynomial(3)
L.<a> = NumberField(P)
Cl_L = L.class_group()
if p.divides( Cl_L.order() ):
print "E =", E
print "RANK ::", E.rank()
print "P =", P
print "L =", L
print "Cl(L) ~", L.class_group()
break
</code></pre>
<p>The above rudimentary search finds:</p>
<pre><code>E = Elliptic Curve defined by y^2 = x^3 - x + 19 over Rational Field
RANK :: 1
P = 3*x^4 - 6*x^2 + 228*x - 1
L = Number Field in a with defining polynomial 3*x^4 - 6*x^2 + 228*x - 1
Cl(L) ~ Class group of order 15 with structure C15 of Number Field in a with defining polynomial 3*x^4 - 6*x^2 + 228*x - 1
</code></pre>
https://ask.sagemath.org/question/42396/class-groups-of-number-fields-obtained-by-adding-p-torsion-points-of-elliptic-curves/?comment=42417#post-id-42417Thank you very much for your response. I will try to do some experiments using the above codes.
The examples that I have in mind are the the two modular curves $X_0(11)$ and $X_0(17)$.Wed, 23 May 2018 20:58:01 -0500https://ask.sagemath.org/question/42396/class-groups-of-number-fields-obtained-by-adding-p-torsion-points-of-elliptic-curves/?comment=42417#post-id-42417