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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 303450bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 303450.bc5 | 303450bc1 | \([1, 1, 0, -507513050, 4394954356500]\) | \(38331145780597164097/55468445663232\) | \(20919897414359580672000000\) | \([2]\) | \(141557760\) | \(3.7607\) | \(\Gamma_0(N)\)-optimal |
| 303450.bc4 | 303450bc2 | \([1, 1, 0, -655481050, 1623069812500]\) | \(82582985847542515777/44772582831427584\) | \(16885957928153104332864000000\) | \([2, 2]\) | \(283115520\) | \(4.1073\) | |
| 303450.bc6 | 303450bc3 | \([1, 1, 0, 2528142950, 12768937436500]\) | \(4738217997934888496063/2928751705237796928\) | \(-1104577287016327886837313000000\) | \([2]\) | \(566231040\) | \(4.4539\) | |
| 303450.bc2 | 303450bc4 | \([1, 1, 0, -6206593050, -186920449267500]\) | \(70108386184777836280897/552468975892674624\) | \(208363406655762036427329000000\) | \([2, 2]\) | \(566231040\) | \(4.4539\) | |
| 303450.bc3 | 303450bc5 | \([1, 1, 0, -2114064050, -429734287366500]\) | \(-2770540998624539614657/209924951154647363208\) | \(-79173093645577037501580646125000\) | \([2]\) | \(1132462080\) | \(4.8005\) | |
| 303450.bc1 | 303450bc6 | \([1, 1, 0, -99116914050, -12010780810048500]\) | \(285531136548675601769470657/17941034271597192\) | \(6766452385344405657910125000\) | \([2]\) | \(1132462080\) | \(4.8005\) |
Rank
sage: E.rank()
The elliptic curves in class 303450bc have rank \(0\).
Complex multiplication
The elliptic curves in class 303450bc do not have complex multiplication.Modular form 303450.2.a.bc
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.
