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SageMath
E = EllipticCurve("es1")
E.isogeny_class()
Elliptic curves in class 55440.es
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55440.es1 | 55440et4 | \([0, 0, 0, -295707, -61892854]\) | \(957681397954009/31185\) | \(93117911040\) | \([2]\) | \(196608\) | \(1.6060\) | |
| 55440.es2 | 55440et3 | \([0, 0, 0, -29307, 290666]\) | \(932288503609/527295615\) | \(1574496269660160\) | \([4]\) | \(196608\) | \(1.6060\) | |
| 55440.es3 | 55440et2 | \([0, 0, 0, -18507, -964294]\) | \(234770924809/1334025\) | \(3983377305600\) | \([2, 2]\) | \(98304\) | \(1.2595\) | |
| 55440.es4 | 55440et1 | \([0, 0, 0, -507, -31894]\) | \(-4826809/144375\) | \(-431101440000\) | \([2]\) | \(49152\) | \(0.91290\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 55440.es have rank \(0\).
Complex multiplication
The elliptic curves in class 55440.es do not have complex multiplication.Modular form 55440.2.a.es
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.
