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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 30912.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 30912.p1 | 30912n4 | \([0, -1, 0, -97055393, -366208360671]\) | \(385693937170561837203625/2159357734550274048\) | \(566062673965947040038912\) | \([2]\) | \(5529600\) | \(3.4001\) | |
| 30912.p2 | 30912n2 | \([0, -1, 0, -7167713, 7039718241]\) | \(155355156733986861625/8291568305839392\) | \(2173584881965961576448\) | \([2]\) | \(1843200\) | \(2.8508\) | |
| 30912.p3 | 30912n3 | \([0, -1, 0, -2683553, -12068593887]\) | \(-8152944444844179625/235342826399858688\) | \(-61693709883764555907072\) | \([2]\) | \(2764800\) | \(3.0535\) | |
| 30912.p4 | 30912n1 | \([0, -1, 0, 297247, 439200609]\) | \(11079872671250375/324440155855872\) | \(-85050040216681709568\) | \([2]\) | \(921600\) | \(2.5042\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 30912.p have rank \(1\).
Complex multiplication
The elliptic curves in class 30912.p do not have complex multiplication.Modular form 30912.2.a.p
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
