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SageMath
E = EllipticCurve("ga1")
E.isogeny_class()
Elliptic curves in class 179520ga
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 179520.fi4 | 179520ga1 | \([0, 1, 0, -467841, 123011295]\) | \(43199583152847841/89760000\) | \(23530045440000\) | \([2]\) | \(1179648\) | \(1.8152\) | \(\Gamma_0(N)\)-optimal |
| 179520.fi3 | 179520ga2 | \([0, 1, 0, -472961, 120175839]\) | \(44633474953947361/1967006250000\) | \(515638886400000000\) | \([2, 2]\) | \(2359296\) | \(2.1617\) | |
| 179520.fi5 | 179520ga3 | \([0, 1, 0, 245119, 453508575]\) | \(6213165856218719/342407226562500\) | \(-89760000000000000000\) | \([2]\) | \(4718592\) | \(2.5083\) | |
| 179520.fi2 | 179520ga4 | \([0, 1, 0, -1272961, -394544161]\) | \(870220733067747361/247623269602500\) | \(64912954386677760000\) | \([2, 2]\) | \(4718592\) | \(2.5083\) | |
| 179520.fi6 | 179520ga5 | \([0, 1, 0, 3351039, -2598342561]\) | \(15875306080318016639/20322604533582450\) | \(-5327448842851437772800\) | \([4]\) | \(9437184\) | \(2.8549\) | |
| 179520.fi1 | 179520ga6 | \([0, 1, 0, -18696961, -31120025761]\) | \(2757381641970898311361/379829992662450\) | \(99570153596505292800\) | \([2]\) | \(9437184\) | \(2.8549\) |
Rank
sage: E.rank()
The elliptic curves in class 179520ga have rank \(2\).
Complex multiplication
The elliptic curves in class 179520ga do not have complex multiplication.Modular form 179520.2.a.ga
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
