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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 5040.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5040.g1 | 5040bc7 | \([0, 0, 0, -50577483, -138447122182]\) | \(4791901410190533590281/41160000\) | \(122903101440000\) | \([2]\) | \(221184\) | \(2.7447\) | |
5040.g2 | 5040bc6 | \([0, 0, 0, -3161163, -2163135238]\) | \(1169975873419524361/108425318400\) | \(323756265937305600\) | \([2, 2]\) | \(110592\) | \(2.3981\) | |
5040.g3 | 5040bc8 | \([0, 0, 0, -2930763, -2491823878]\) | \(-932348627918877961/358766164249920\) | \(-1071270026191633121280\) | \([2]\) | \(221184\) | \(2.7447\) | |
5040.g4 | 5040bc4 | \([0, 0, 0, -627483, -187952182]\) | \(9150443179640281/184570312500\) | \(551124000000000000\) | \([2]\) | \(73728\) | \(2.1954\) | |
5040.g5 | 5040bc3 | \([0, 0, 0, -212043, -28562182]\) | \(353108405631241/86318776320\) | \(257746484991098880\) | \([2]\) | \(55296\) | \(2.0516\) | |
5040.g6 | 5040bc2 | \([0, 0, 0, -83163, 4845962]\) | \(21302308926361/8930250000\) | \(26665583616000000\) | \([2, 2]\) | \(36864\) | \(1.8488\) | |
5040.g7 | 5040bc1 | \([0, 0, 0, -71643, 7378058]\) | \(13619385906841/6048000\) | \(18059231232000\) | \([2]\) | \(18432\) | \(1.5023\) | \(\Gamma_0(N)\)-optimal |
5040.g8 | 5040bc5 | \([0, 0, 0, 276837, 35589962]\) | \(785793873833639/637994920500\) | \(-1905042624694272000\) | \([2]\) | \(73728\) | \(2.1954\) |
Rank
sage: E.rank()
The elliptic curves in class 5040.g have rank \(1\).
Complex multiplication
The elliptic curves in class 5040.g do not have complex multiplication.Modular form 5040.2.a.g
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.