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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 6630.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6630.q1 | 6630t3 | \([1, 1, 1, -377310, -89363493]\) | \(5940441603429810927841/3044264109120\) | \(3044264109120\) | \([2]\) | \(73728\) | \(1.7276\) | |
| 6630.q2 | 6630t2 | \([1, 1, 1, -23710, -1387813]\) | \(1474074790091785441/32813650022400\) | \(32813650022400\) | \([2, 2]\) | \(36864\) | \(1.3810\) | |
| 6630.q3 | 6630t1 | \([1, 1, 1, -3230, 37595]\) | \(3726830856733921/1501644718080\) | \(1501644718080\) | \([4]\) | \(18432\) | \(1.0344\) | \(\Gamma_0(N)\)-optimal |
| 6630.q4 | 6630t4 | \([1, 1, 1, 2210, -4228645]\) | \(1193680917131039/7728836230440000\) | \(-7728836230440000\) | \([4]\) | \(73728\) | \(1.7276\) |
Rank
sage: E.rank()
The elliptic curves in class 6630.q have rank \(1\).
Complex multiplication
The elliptic curves in class 6630.q do not have complex multiplication.Modular form 6630.2.a.q
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
