Properties

Label 12635.e
Number of curves $3$
Conductor $12635$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands for: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 12635.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12635.e1 12635a3 \([0, -1, 1, -47411, 4172421]\) \(-250523582464/13671875\) \(-643205404296875\) \([]\) \(42768\) \(1.5997\)  
12635.e2 12635a1 \([0, -1, 1, -481, -4349]\) \(-262144/35\) \(-1646605835\) \([]\) \(4752\) \(0.50107\) \(\Gamma_0(N)\)-optimal
12635.e3 12635a2 \([0, -1, 1, 3129, 10452]\) \(71991296/42875\) \(-2017092147875\) \([]\) \(14256\) \(1.0504\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12635.e have rank \(1\).

Complex multiplication

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

Modular form 12635.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2q^{4} - q^{5} + q^{7} - 2q^{9} - 3q^{11} + 2q^{12} - 5q^{13} + q^{15} + 4q^{16} + 3q^{17} + O(q^{20})\)  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 LMFDB numbering.

\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.