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SageMath
E = EllipticCurve("de1")
E.isogeny_class()
Elliptic curves in class 199920.de
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 199920.de1 | 199920cp4 | \([0, -1, 0, -47981600, -127910414400]\) | \(25351269426118370449/27551475\) | \(13276788663398400\) | \([2]\) | \(9437184\) | \(2.8125\) | |
| 199920.de2 | 199920cp3 | \([0, -1, 0, -3740480, -934081728]\) | \(12010404962647729/6166198828125\) | \(2971431427809600000000\) | \([4]\) | \(9437184\) | \(2.8125\) | |
| 199920.de3 | 199920cp2 | \([0, -1, 0, -2999600, -1996800000]\) | \(6193921595708449/6452105625\) | \(3109207141071360000\) | \([2, 2]\) | \(4718592\) | \(2.4659\) | |
| 199920.de4 | 199920cp1 | \([0, -1, 0, -141920, -46719168]\) | \(-656008386769/1581036975\) | \(-761886388517990400\) | \([2]\) | \(2359296\) | \(2.1193\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 199920.de have rank \(1\).
Complex multiplication
The elliptic curves in class 199920.de do not have complex multiplication.Modular form 199920.2.a.de
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
