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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 6630ba
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6630.z4 | 6630ba1 | \([1, 0, 0, -305, -423]\) | \(3138428376721/1747933200\) | \(1747933200\) | \([4]\) | \(3072\) | \(0.46381\) | \(\Gamma_0(N)\)-optimal |
| 6630.z2 | 6630ba2 | \([1, 0, 0, -3685, -86275]\) | \(5534056064805841/9890302500\) | \(9890302500\) | \([2, 2]\) | \(6144\) | \(0.81038\) | |
| 6630.z1 | 6630ba3 | \([1, 0, 0, -58935, -5511825]\) | \(22638311752145721841/72499050\) | \(72499050\) | \([2]\) | \(12288\) | \(1.1570\) | |
| 6630.z3 | 6630ba4 | \([1, 0, 0, -2515, -141733]\) | \(-1759334717565361/7634341406250\) | \(-7634341406250\) | \([2]\) | \(12288\) | \(1.1570\) |
Rank
sage: E.rank()
The elliptic curves in class 6630ba have rank \(0\).
Complex multiplication
The elliptic curves in class 6630ba do not have complex multiplication.Modular form 6630.2.a.ba
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.
