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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 27075.s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 27075.s1 | 27075q4 | \([1, 0, 1, -916226, -337633027]\) | \(115714886617/1539\) | \(1131306419671875\) | \([2]\) | \(276480\) | \(2.0319\) | |
| 27075.s2 | 27075q2 | \([1, 0, 1, -58851, -4971527]\) | \(30664297/3249\) | \(2388313552640625\) | \([2, 2]\) | \(138240\) | \(1.6854\) | |
| 27075.s3 | 27075q1 | \([1, 0, 1, -13726, 533723]\) | \(389017/57\) | \(41900237765625\) | \([2]\) | \(69120\) | \(1.3388\) | \(\Gamma_0(N)\)-optimal |
| 27075.s4 | 27075q3 | \([1, 0, 1, 76524, -24465527]\) | \(67419143/390963\) | \(-287393730834421875\) | \([2]\) | \(276480\) | \(2.0319\) |
Rank
sage: E.rank()
The elliptic curves in class 27075.s have rank \(1\).
Complex multiplication
The elliptic curves in class 27075.s do not have complex multiplication.Modular form 27075.2.a.s
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.
