Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 111720d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 111720.e3 | 111720d1 | \([0, -1, 0, -12511, -241100]\) | \(115060504576/52780005\) | \(99352236931920\) | \([2]\) | \(368640\) | \(1.3806\) | \(\Gamma_0(N)\)-optimal |
| 111720.e2 | 111720d2 | \([0, -1, 0, -100956, 12211956]\) | \(3778298043856/59213025\) | \(1783386413625600\) | \([2, 2]\) | \(737280\) | \(1.7272\) | |
| 111720.e4 | 111720d3 | \([0, -1, 0, -7856, 33773916]\) | \(-445138564/4089438495\) | \(-492665189886213120\) | \([2]\) | \(1474560\) | \(2.0738\) | |
| 111720.e1 | 111720d4 | \([0, -1, 0, -1609176, 786230460]\) | \(3825131988299044/961875\) | \(115879559040000\) | \([2]\) | \(1474560\) | \(2.0738\) |
Rank
sage: E.rank()
The elliptic curves in class 111720d have rank \(0\).
Complex multiplication
The elliptic curves in class 111720d do not have complex multiplication.Modular form 111720.2.a.d
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.
