Properties

Label 8712y
Number of curves $6$
Conductor $8712$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("y1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 8712y

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8712.u5 8712y1 \([0, 0, 0, 726, 9317]\) \(2048/3\) \(-61990462512\) \([2]\) \(5120\) \(0.75633\) \(\Gamma_0(N)\)-optimal
8712.u4 8712y2 \([0, 0, 0, -4719, 93170]\) \(35152/9\) \(2975542200576\) \([2, 2]\) \(10240\) \(1.1029\)  
8712.u3 8712y3 \([0, 0, 0, -26499, -1583890]\) \(1556068/81\) \(107119519220736\) \([2, 2]\) \(20480\) \(1.4495\)  
8712.u2 8712y4 \([0, 0, 0, -70059, 7136822]\) \(28756228/3\) \(3967389600768\) \([2]\) \(20480\) \(1.4495\)  
8712.u1 8712y5 \([0, 0, 0, -418539, -104219962]\) \(3065617154/9\) \(23804337604608\) \([2]\) \(40960\) \(1.7960\)  
8712.u6 8712y6 \([0, 0, 0, 17061, -6279658]\) \(207646/6561\) \(-17353362113759232\) \([2]\) \(40960\) \(1.7960\)  

Rank

sage: E.rank()
 

The elliptic curves in class 8712y have rank \(0\).

Complex multiplication

The elliptic curves in class 8712y do not have complex multiplication.

Modular form 8712.2.a.y

sage: E.q_eigenform(10)
 
\(q + 2q^{5} + 2q^{13} + 2q^{17} + 4q^{19} + O(q^{20})\)  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 Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.