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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 6930.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6930.a1 | 6930g5 | \([1, -1, 0, -426105, 107150665]\) | \(11736717412386894481/1890645330420\) | \(1378280445876180\) | \([2]\) | \(65536\) | \(1.9145\) | |
| 6930.a2 | 6930g3 | \([1, -1, 0, -177525, -28745339]\) | \(848742840525560401/1443750000\) | \(1052493750000\) | \([2]\) | \(32768\) | \(1.5679\) | |
| 6930.a3 | 6930g4 | \([1, -1, 0, -29205, 1337125]\) | \(3778993806976081/1138958528400\) | \(830300767203600\) | \([2, 2]\) | \(32768\) | \(1.5679\) | |
| 6930.a4 | 6930g2 | \([1, -1, 0, -11205, -437675]\) | \(213429068128081/8537760000\) | \(6224027040000\) | \([2, 2]\) | \(16384\) | \(1.2214\) | |
| 6930.a5 | 6930g1 | \([1, -1, 0, 315, -25259]\) | \(4733169839/378470400\) | \(-275904921600\) | \([2]\) | \(8192\) | \(0.87479\) | \(\Gamma_0(N)\)-optimal |
| 6930.a6 | 6930g6 | \([1, -1, 0, 79695, 8894785]\) | \(76786760064334319/91531319653620\) | \(-66726332027488980\) | \([2]\) | \(65536\) | \(1.9145\) |
Rank
sage: E.rank()
The elliptic curves in class 6930.a have rank \(1\).
Complex multiplication
The elliptic curves in class 6930.a do not have complex multiplication.Modular form 6930.2.a.a
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
