Properties

Label 75712.cn
Number of curves $2$
Conductor $75712$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("cn1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 75712.cn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
75712.cn1 75712cx2 \([0, -1, 0, -27265, 1721121]\) \(3543122/49\) \(31000315953152\) \([2]\) \(301056\) \(1.3944\)  
75712.cn2 75712cx1 \([0, -1, 0, -225, 71681]\) \(-4/7\) \(-2214308282368\) \([2]\) \(150528\) \(1.0478\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 75712.cn have rank \(1\).

Complex multiplication

The elliptic curves in class 75712.cn do not have complex multiplication.

Modular form 75712.2.a.cn

sage: E.q_eigenform(10)
 
\(q + 2q^{3} - 4q^{5} + q^{7} + q^{9} - 8q^{15} - 2q^{17} + 2q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.