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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 273999f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 273999.f6 | 273999f1 | \([1, 1, 0, 11184, 1339299]\) | \(3288008303/18259263\) | \(-859023114245703\) | \([2]\) | \(884736\) | \(1.5476\) | \(\Gamma_0(N)\)-optimal |
| 273999.f5 | 273999f2 | \([1, 1, 0, -135021, 17158680]\) | \(5786435182177/627352209\) | \(29514337369701129\) | \([2, 2]\) | \(1769472\) | \(1.8942\) | |
| 273999.f2 | 273999f3 | \([1, 1, 0, -2100666, 1170992295]\) | \(21790813729717297/304746849\) | \(14337083993178969\) | \([2, 2]\) | \(3538944\) | \(2.2408\) | |
| 273999.f4 | 273999f4 | \([1, 1, 0, -508656, -121310451]\) | \(309368403125137/44372288367\) | \(2087533398211566327\) | \([2]\) | \(3538944\) | \(2.2408\) | |
| 273999.f1 | 273999f5 | \([1, 1, 0, -33610551, 74986048896]\) | \(89254274298475942657/17457\) | \(821279944617\) | \([2]\) | \(7077888\) | \(2.5873\) | |
| 273999.f3 | 273999f6 | \([1, 1, 0, -2041101, 1240599954]\) | \(-19989223566735457/2584262514273\) | \(-121578906719248359513\) | \([2]\) | \(7077888\) | \(2.5873\) |
Rank
sage: E.rank()
The elliptic curves in class 273999f have rank \(1\).
Complex multiplication
The elliptic curves in class 273999f do not have complex multiplication.Modular form 273999.2.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.
