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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 121242w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121242.u4 | 121242w1 | \([1, 1, 1, -27409, 5160431]\) | \(-1285429208617/5778702336\) | \(-10237323689066496\) | \([4]\) | \(806400\) | \(1.7569\) | \(\Gamma_0(N)\)-optimal |
| 121242.u3 | 121242w2 | \([1, 1, 1, -646929, 199689711]\) | \(16901976846788137/31100027904\) | \(55095596533638144\) | \([2, 2]\) | \(1612800\) | \(2.1035\) | |
| 121242.u2 | 121242w3 | \([1, 1, 1, -859889, 56665775]\) | \(39691253323129897/22176528704352\) | \(39287073368010533472\) | \([2]\) | \(3225600\) | \(2.4501\) | |
| 121242.u1 | 121242w4 | \([1, 1, 1, -10346289, 12804977967]\) | \(69138733474448992297/234724512\) | \(415828791203232\) | \([2]\) | \(3225600\) | \(2.4501\) |
Rank
sage: E.rank()
The elliptic curves in class 121242w have rank \(1\).
Complex multiplication
The elliptic curves in class 121242w do not have complex multiplication.Modular form 121242.2.a.w
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
