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SageMath
E = EllipticCurve("em1")
E.isogeny_class()
Elliptic curves in class 92400em
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
92400.dv3 | 92400em1 | \([0, -1, 0, -6497008, 6376256512]\) | \(473897054735271721/779625\) | \(49896000000000\) | \([2]\) | \(1769472\) | \(2.3202\) | \(\Gamma_0(N)\)-optimal |
92400.dv2 | 92400em2 | \([0, -1, 0, -6499008, 6372136512]\) | \(474334834335054841/607815140625\) | \(38900169000000000000\) | \([2, 2]\) | \(3538944\) | \(2.6668\) | |
92400.dv4 | 92400em3 | \([0, -1, 0, -4749008, 9879136512]\) | \(-185077034913624841/551466161890875\) | \(-35293834361016000000000\) | \([2]\) | \(7077888\) | \(3.0133\) | |
92400.dv1 | 92400em4 | \([0, -1, 0, -8281008, 2601424512]\) | \(981281029968144361/522287841796875\) | \(33426421875000000000000\) | \([2]\) | \(7077888\) | \(3.0133\) |
Rank
sage: E.rank()
The elliptic curves in class 92400em have rank \(0\).
Complex multiplication
The elliptic curves in class 92400em do not have complex multiplication.Modular form 92400.2.a.em
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.