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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 485520cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 485520.cd3 | 485520cd1 | \([0, -1, 0, -1549136, 742648320]\) | \(4158523459441/16065\) | \(1588306108354560\) | \([2]\) | \(7077888\) | \(2.1303\) | \(\Gamma_0(N)\)-optimal |
| 485520.cd2 | 485520cd2 | \([0, -1, 0, -1572256, 719361856]\) | \(4347507044161/258084225\) | \(25516137630716006400\) | \([2, 2]\) | \(14155776\) | \(2.4768\) | |
| 485520.cd4 | 485520cd3 | \([0, -1, 0, 1179024, 2966607360]\) | \(1833318007919/39525924375\) | \(-3907828641342850560000\) | \([2]\) | \(28311552\) | \(2.8234\) | |
| 485520.cd1 | 485520cd4 | \([0, -1, 0, -4693456, -3018587264]\) | \(115650783909361/27072079335\) | \(2676548333248661975040\) | \([2]\) | \(28311552\) | \(2.8234\) |
Rank
sage: E.rank()
The elliptic curves in class 485520cd have rank \(1\).
Complex multiplication
The elliptic curves in class 485520cd do not have complex multiplication.Modular form 485520.2.a.cd
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.
