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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 30912j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 30912.k4 | 30912j1 | \([0, -1, 0, 1071, -10287]\) | \(8284506032/7394247\) | \(-121147342848\) | \([2]\) | \(32768\) | \(0.81410\) | \(\Gamma_0(N)\)-optimal |
| 30912.k3 | 30912j2 | \([0, -1, 0, -5409, -86751]\) | \(267100692772/102880449\) | \(6742373105664\) | \([2, 2]\) | \(65536\) | \(1.1607\) | |
| 30912.k2 | 30912j3 | \([0, -1, 0, -38529, 2860929]\) | \(48260105780546/1193313807\) | \(156410027311104\) | \([4]\) | \(131072\) | \(1.5072\) | |
| 30912.k1 | 30912j4 | \([0, -1, 0, -75969, -8031807]\) | \(369937818893666/123409881\) | \(16175579922432\) | \([2]\) | \(131072\) | \(1.5072\) |
Rank
sage: E.rank()
The elliptic curves in class 30912j have rank \(2\).
Complex multiplication
The elliptic curves in class 30912j do not have complex multiplication.Modular form 30912.2.a.j
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.
