Properties

Label 5712.e
Number of curves $4$
Conductor $5712$
CM no
Rank $0$
Graph

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

Elliptic curves in class 5712.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5712.e1 5712b3 \([0, -1, 0, -544, 4624]\) \(17418812548/1753941\) \(1796035584\) \([4]\) \(2560\) \(0.51187\)  
5712.e2 5712b2 \([0, -1, 0, -124, -416]\) \(830321872/127449\) \(32626944\) \([2, 2]\) \(1280\) \(0.16530\)  
5712.e3 5712b1 \([0, -1, 0, -119, -462]\) \(11745974272/357\) \(5712\) \([2]\) \(640\) \(-0.18127\) \(\Gamma_0(N)\)-optimal
5712.e4 5712b4 \([0, -1, 0, 216, -2592]\) \(1083360092/3306177\) \(-3385525248\) \([2]\) \(2560\) \(0.51187\)  

Rank

sage: E.rank()
 

The elliptic curves in class 5712.e have rank \(0\).

Complex multiplication

The elliptic curves in class 5712.e do not have complex multiplication.

Modular form 5712.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} - q^{7} + q^{9} + 4 q^{11} + 6 q^{13} + 2 q^{15} + q^{17} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.