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SageMath
E = EllipticCurve("cv1")
E.isogeny_class()
Elliptic curves in class 112710cv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
112710.cl4 | 112710cv1 | \([1, 0, 0, 456614, 20175716]\) | \(436192097814719/259683840000\) | \(-6268136606184960000\) | \([2]\) | \(3317760\) | \(2.2962\) | \(\Gamma_0(N)\)-optimal |
112710.cl3 | 112710cv2 | \([1, 0, 0, -1855386, 162132516]\) | \(29263955267177281/16463793153600\) | \(397395943246747598400\) | \([2, 2]\) | \(6635520\) | \(2.6427\) | |
112710.cl2 | 112710cv3 | \([1, 0, 0, -18559586, -30617026404]\) | \(29291056630578924481/175463302795560\) | \(4235257578195722393640\) | \([2]\) | \(13271040\) | \(2.9893\) | |
112710.cl1 | 112710cv4 | \([1, 0, 0, -22143186, 40035774636]\) | \(49745123032831462081/97939634471640\) | \(2364024684893989043160\) | \([2]\) | \(13271040\) | \(2.9893\) |
Rank
sage: E.rank()
The elliptic curves in class 112710cv have rank \(0\).
Complex multiplication
The elliptic curves in class 112710cv do not have complex multiplication.Modular form 112710.2.a.cv
sage: E.q_eigenform(10)
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.