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Properties

Label 12635h
Number of curves $2$
Conductor $12635$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("h1")
 
E.isogeny_class()
 

Elliptic curves in class 12635h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12635.i2 12635h1 \([0, 1, 1, -75930, -46173869]\) \(-1029077364736/18960396875\) \(-892008575094021875\) \([]\) \(216000\) \(2.1254\) \(\Gamma_0(N)\)-optimal
12635.i1 12635h2 \([0, 1, 1, -6014380, 7454518071]\) \(-511416541770305536/214587319023035\) \(-10095449474866740868835\) \([]\) \(1080000\) \(2.9301\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12635h have rank \(0\).

Complex multiplication

The elliptic curves in class 12635h do not have complex multiplication.

Modular form 12635.2.a.h

sage: E.q_eigenform(10)
 
\(q + 2 q^{2} + q^{3} + 2 q^{4} + q^{5} + 2 q^{6} + q^{7} - 2 q^{9} + 2 q^{10} - 3 q^{11} + 2 q^{12} + q^{13} + 2 q^{14} + q^{15} - 4 q^{16} + 3 q^{17} - 4 q^{18} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.