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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 6720.bi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6720.bi1 | 6720bw7 | \([0, 1, 0, -22478881, 41013876575]\) | \(4791901410190533590281/41160000\) | \(10789847040000\) | \([2]\) | \(221184\) | \(2.5420\) | |
6720.bi2 | 6720bw6 | \([0, 1, 0, -1404961, 640460639]\) | \(1169975873419524361/108425318400\) | \(28423046666649600\) | \([2, 2]\) | \(110592\) | \(2.1954\) | |
6720.bi3 | 6720bw8 | \([0, 1, 0, -1302561, 737883999]\) | \(-932348627918877961/358766164249920\) | \(-94048397361131028480\) | \([2]\) | \(221184\) | \(2.5420\) | |
6720.bi4 | 6720bw4 | \([0, 1, 0, -278881, 55596575]\) | \(9150443179640281/184570312500\) | \(48384000000000000\) | \([2]\) | \(73728\) | \(1.9927\) | |
6720.bi5 | 6720bw3 | \([0, 1, 0, -94241, 8431455]\) | \(353108405631241/86318776320\) | \(22627949299630080\) | \([2]\) | \(55296\) | \(1.8488\) | |
6720.bi6 | 6720bw2 | \([0, 1, 0, -36961, -1448161]\) | \(21302308926361/8930250000\) | \(2341011456000000\) | \([2, 2]\) | \(36864\) | \(1.6461\) | |
6720.bi7 | 6720bw1 | \([0, 1, 0, -31841, -2196705]\) | \(13619385906841/6048000\) | \(1585446912000\) | \([2]\) | \(18432\) | \(1.2995\) | \(\Gamma_0(N)\)-optimal |
6720.bi8 | 6720bw5 | \([0, 1, 0, 123039, -10504161]\) | \(785793873833639/637994920500\) | \(-167246540439552000\) | \([2]\) | \(73728\) | \(1.9927\) |
Rank
sage: E.rank()
The elliptic curves in class 6720.bi have rank \(1\).
Complex multiplication
The elliptic curves in class 6720.bi do not have complex multiplication.Modular form 6720.2.a.bi
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.