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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 17328f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17328.d5 | 17328f1 | \([0, -1, 0, 241, 1698]\) | \(2048/3\) | \(-2258202288\) | \([2]\) | \(6912\) | \(0.48029\) | \(\Gamma_0(N)\)-optimal |
| 17328.d4 | 17328f2 | \([0, -1, 0, -1564, 18304]\) | \(35152/9\) | \(108393709824\) | \([2, 2]\) | \(13824\) | \(0.82687\) | |
| 17328.d3 | 17328f3 | \([0, -1, 0, -8784, -299376]\) | \(1556068/81\) | \(3902173553664\) | \([2, 2]\) | \(27648\) | \(1.1734\) | |
| 17328.d2 | 17328f4 | \([0, -1, 0, -23224, 1369888]\) | \(28756228/3\) | \(144524946432\) | \([2]\) | \(27648\) | \(1.1734\) | |
| 17328.d1 | 17328f5 | \([0, -1, 0, -138744, -19845360]\) | \(3065617154/9\) | \(867149678592\) | \([2]\) | \(55296\) | \(1.5200\) | |
| 17328.d6 | 17328f6 | \([0, -1, 0, 5656, -1200432]\) | \(207646/6561\) | \(-632152115693568\) | \([2]\) | \(55296\) | \(1.5200\) |
Rank
sage: E.rank()
The elliptic curves in class 17328f have rank \(0\).
Complex multiplication
The elliptic curves in class 17328f do not have complex multiplication.Modular form 17328.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
