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SageMath
E = EllipticCurve("bv1")
E.isogeny_class()
Elliptic curves in class 479370bv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
479370.bv4 | 479370bv1 | \([1, 1, 1, 3091919, -12558074161]\) | \(5495662324535111/117739817533440\) | \(-70034389279174809354240\) | \([2]\) | \(50176000\) | \(3.0639\) | \(\Gamma_0(N)\)-optimal* |
479370.bv3 | 479370bv2 | \([1, 1, 1, -65802801, -194633040177]\) | \(52974743974734147769/3152005008998400\) | \(1874886087261063171686400\) | \([2, 2]\) | \(100352000\) | \(3.4105\) | \(\Gamma_0(N)\)-optimal* |
479370.bv2 | 479370bv3 | \([1, 1, 1, -196595121, 819112073679]\) | \(1412712966892699019449/330160465517040000\) | \(196387144561751714889840000\) | \([2]\) | \(200704000\) | \(3.7571\) | \(\Gamma_0(N)\)-optimal* |
479370.bv1 | 479370bv4 | \([1, 1, 1, -1037326001, -12859798084657]\) | \(207530301091125281552569/805586668007040\) | \(479181737217271984179840\) | \([2]\) | \(200704000\) | \(3.7571\) |
Rank
sage: E.rank()
The elliptic curves in class 479370bv have rank \(1\).
Complex multiplication
The elliptic curves in class 479370bv do not have complex multiplication.Modular form 479370.2.a.bv
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.