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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 130130c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 130130.c3 | 130130c1 | \([1, 0, 1, -370114, -78502188]\) | \(1161631688686561/121121000000\) | \(584627932889000000\) | \([2]\) | \(2419200\) | \(2.1442\) | \(\Gamma_0(N)\)-optimal |
| 130130.c4 | 130130c2 | \([1, 0, 1, 474886, -384730188]\) | \(2453765252833439/14670296641000\) | \(-70810719859448569000\) | \([2]\) | \(4838400\) | \(2.4908\) | |
| 130130.c1 | 130130c3 | \([1, 0, 1, -29184614, -60687107388]\) | \(569541582763202518561/828928100\) | \(4001077613432900\) | \([2]\) | \(7257600\) | \(2.6935\) | |
| 130130.c2 | 130130c4 | \([1, 0, 1, -29176164, -60724003468]\) | \(-569047017391330383361/687121794969610\) | \(-3316605664055468274490\) | \([2]\) | \(14515200\) | \(3.0401\) |
Rank
sage: E.rank()
The elliptic curves in class 130130c have rank \(1\).
Complex multiplication
The elliptic curves in class 130130c do not have complex multiplication.Modular form 130130.2.a.c
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
