# 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 $L=\mathbb{Q}(E[p])$.

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Please 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

more

Thank 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 $X_0(17)$.

( 2018-05-23 20:58:01 -0500 )edit

My 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.

( 2018-05-23 23:05:58 -0500 )edit