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SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 283920.bl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
283920.bl1 | 283920bl7 | \([0, -1, 0, -949732736, 11265810186240]\) | \(4791901410190533590281/41160000\) | \(813758293770240000\) | \([2]\) | \(63700992\) | \(3.4779\) | |
283920.bl2 | 283920bl6 | \([0, -1, 0, -59359616, 176034902016]\) | \(1169975873419524361/108425318400\) | \(2143634647781317017600\) | \([2, 2]\) | \(31850496\) | \(3.1313\) | |
283920.bl3 | 283920bl8 | \([0, -1, 0, -55033216, 202778976256]\) | \(-932348627918877961/358766164249920\) | \(-7093025794035679663226880\) | \([2]\) | \(63700992\) | \(3.4779\) | |
283920.bl4 | 283920bl4 | \([0, -1, 0, -11782736, 15297666240]\) | \(9150443179640281/184570312500\) | \(3649067604000000000000\) | \([2]\) | \(21233664\) | \(2.9286\) | |
283920.bl5 | 283920bl3 | \([0, -1, 0, -3981696, 2325442560]\) | \(353108405631241/86318776320\) | \(1706574833296846356480\) | \([2]\) | \(15925248\) | \(2.7847\) | |
283920.bl6 | 283920bl2 | \([0, -1, 0, -1561616, -393797184]\) | \(21302308926361/8930250000\) | \(176556486951936000000\) | \([2, 2]\) | \(10616832\) | \(2.5820\) | |
283920.bl7 | 283920bl1 | \([0, -1, 0, -1345296, -599906880]\) | \(13619385906841/6048000\) | \(119572647247872000\) | \([2]\) | \(5308416\) | \(2.2354\) | \(\Gamma_0(N)\)-optimal |
283920.bl8 | 283920bl5 | \([0, -1, 0, 5198384, -2897701184]\) | \(785793873833639/637994920500\) | \(-12613548540820211712000\) | \([2]\) | \(21233664\) | \(2.9286\) |
Rank
sage: E.rank()
The elliptic curves in class 283920.bl have rank \(1\).
Complex multiplication
The elliptic curves in class 283920.bl do not have complex multiplication.Modular form 283920.2.a.bl
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.