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SageMath
E = EllipticCurve("gp1")
E.isogeny_class()
Elliptic curves in class 303450.gp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 303450.gp1 | 303450gp8 | \([1, 0, 0, -2537654963, -49203821951583]\) | \(4791901410190533590281/41160000\) | \(15523474063125000000\) | \([2]\) | \(127401984\) | \(3.7236\) | |
| 303450.gp2 | 303450gp6 | \([1, 0, 0, -158606963, -768783719583]\) | \(1169975873419524361/108425318400\) | \(40892556316046400000000\) | \([2, 2]\) | \(63700992\) | \(3.3770\) | |
| 303450.gp3 | 303450gp7 | \([1, 0, 0, -147046963, -885597519583]\) | \(-932348627918877961/358766164249920\) | \(-135308485069496519445000000\) | \([2]\) | \(127401984\) | \(3.7236\) | |
| 303450.gp4 | 303450gp5 | \([1, 0, 0, -31483088, -66800154708]\) | \(9150443179640281/184570312500\) | \(69610603958129882812500\) | \([2]\) | \(42467328\) | \(3.1743\) | |
| 303450.gp5 | 303450gp3 | \([1, 0, 0, -10638963, -10151783583]\) | \(353108405631241/86318776320\) | \(32555084678430720000000\) | \([2]\) | \(31850496\) | \(3.0304\) | |
| 303450.gp6 | 303450gp2 | \([1, 0, 0, -4172588, 1721889792]\) | \(21302308926361/8930250000\) | \(3368039461910156250000\) | \([2, 2]\) | \(21233664\) | \(2.8277\) | |
| 303450.gp7 | 303450gp1 | \([1, 0, 0, -3594588, 2621835792]\) | \(13619385906841/6048000\) | \(2281000270500000000\) | \([2]\) | \(10616832\) | \(2.4811\) | \(\Gamma_0(N)\)-optimal |
| 303450.gp8 | 303450gp4 | \([1, 0, 0, 13889912, 12649702292]\) | \(785793873833639/637994920500\) | \(-240619475237785382812500\) | \([2]\) | \(42467328\) | \(3.1743\) |
Rank
sage: E.rank()
The elliptic curves in class 303450.gp have rank \(0\).
Complex multiplication
The elliptic curves in class 303450.gp do not have complex multiplication.Modular form 303450.2.a.gp
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
