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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 129285.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129285.f1 | 129285w4 | \([1, -1, 1, -1590998, 772729962]\) | \(126574061279329/16286595\) | \(57308354544183795\) | \([2]\) | \(2408448\) | \(2.2367\) | |
| 129285.f2 | 129285w2 | \([1, -1, 1, -108023, 9887622]\) | \(39616946929/10989225\) | \(38668266906975225\) | \([2, 2]\) | \(1204224\) | \(1.8902\) | |
| 129285.f3 | 129285w1 | \([1, -1, 1, -39578, -2897904]\) | \(1948441249/89505\) | \(314945160328305\) | \([2]\) | \(602112\) | \(1.5436\) | \(\Gamma_0(N)\)-optimal |
| 129285.f4 | 129285w3 | \([1, -1, 1, 279832, 64497606]\) | \(688699320191/910381875\) | \(-3203400542783731875\) | \([2]\) | \(2408448\) | \(2.2367\) |
Rank
sage: E.rank()
The elliptic curves in class 129285.f have rank \(1\).
Complex multiplication
The elliptic curves in class 129285.f do not have complex multiplication.Modular form 129285.2.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 LMFDB 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 LMFDB labels.
