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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 30926.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 30926.a1 | 30926f6 | \([1, 0, 1, -6031721, 5701270380]\) | \(2251439055699625/25088\) | \(270428954173952\) | \([2]\) | \(596160\) | \(2.3382\) | |
| 30926.a2 | 30926f5 | \([1, 0, 1, -376681, 89208684]\) | \(-548347731625/1835008\) | \(-19779946362437632\) | \([2]\) | \(298080\) | \(1.9916\) | |
| 30926.a3 | 30926f4 | \([1, 0, 1, -78466, 6927852]\) | \(4956477625/941192\) | \(10145311233932168\) | \([2]\) | \(198720\) | \(1.7889\) | |
| 30926.a4 | 30926f2 | \([1, 0, 1, -23241, -1364734]\) | \(128787625/98\) | \(1056363102242\) | \([2]\) | \(66240\) | \(1.2396\) | |
| 30926.a5 | 30926f1 | \([1, 0, 1, -1151, -30498]\) | \(-15625/28\) | \(-301818029212\) | \([2]\) | \(33120\) | \(0.89299\) | \(\Gamma_0(N)\)-optimal |
| 30926.a6 | 30926f3 | \([1, 0, 1, 9894, 636620]\) | \(9938375/21952\) | \(-236625334902208\) | \([2]\) | \(99360\) | \(1.4423\) |
Rank
sage: E.rank()
The elliptic curves in class 30926.a have rank \(1\).
Complex multiplication
The elliptic curves in class 30926.a do not have complex multiplication.Modular form 30926.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
