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SageMath
E = EllipticCurve("gm1")
E.isogeny_class()
Elliptic curves in class 78400.gm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 78400.gm1 | 78400u4 | \([0, 0, 0, -1465100, -682570000]\) | \(1443468546/7\) | \(1686616064000000\) | \([2]\) | \(786432\) | \(2.1222\) | |
| 78400.gm2 | 78400u3 | \([0, 0, 0, -289100, 47334000]\) | \(11090466/2401\) | \(578509309952000000\) | \([2]\) | \(786432\) | \(2.1222\) | |
| 78400.gm3 | 78400u2 | \([0, 0, 0, -93100, -10290000]\) | \(740772/49\) | \(5903156224000000\) | \([2, 2]\) | \(393216\) | \(1.7756\) | |
| 78400.gm4 | 78400u1 | \([0, 0, 0, 4900, -686000]\) | \(432/7\) | \(-210827008000000\) | \([2]\) | \(196608\) | \(1.4291\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 78400.gm have rank \(0\).
Complex multiplication
The elliptic curves in class 78400.gm do not have complex multiplication.Modular form 78400.2.a.gm
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.
