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SageMath
E = EllipticCurve("ez1")
E.isogeny_class()
Elliptic curves in class 123840.ez
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 123840.ez1 | 123840gf4 | \([0, 0, 0, -286572, -1887536]\) | \(54477543627364/31494140625\) | \(1504656000000000000\) | \([2]\) | \(1179648\) | \(2.1774\) | |
| 123840.ez2 | 123840gf2 | \([0, 0, 0, -193692, 32700976]\) | \(67283921459536/260015625\) | \(3105609984000000\) | \([2, 2]\) | \(589824\) | \(1.8308\) | |
| 123840.ez3 | 123840gf1 | \([0, 0, 0, -193512, 32764984]\) | \(1073544204384256/16125\) | \(12037248000\) | \([2]\) | \(294912\) | \(1.4843\) | \(\Gamma_0(N)\)-optimal |
| 123840.ez4 | 123840gf3 | \([0, 0, 0, -103692, 63192976]\) | \(-2580786074884/34615360125\) | \(-1653774583799808000\) | \([2]\) | \(1179648\) | \(2.1774\) |
Rank
sage: E.rank()
The elliptic curves in class 123840.ez have rank \(1\).
Complex multiplication
The elliptic curves in class 123840.ez do not have complex multiplication.Modular form 123840.2.a.ez
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 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.
