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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 6840.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6840.g1 | 6840r3 | \([0, 0, 0, -273603, -55084498]\) | \(3034301922374404/1425\) | \(1063756800\) | \([2]\) | \(16384\) | \(1.5058\) | |
6840.g2 | 6840r4 | \([0, 0, 0, -20523, -491722]\) | \(1280615525284/601171875\) | \(448772400000000\) | \([2]\) | \(16384\) | \(1.5058\) | |
6840.g3 | 6840r2 | \([0, 0, 0, -17103, -860398]\) | \(2964647793616/2030625\) | \(378963360000\) | \([2, 2]\) | \(8192\) | \(1.1592\) | |
6840.g4 | 6840r1 | \([0, 0, 0, -858, -18907]\) | \(-5988775936/9774075\) | \(-114004810800\) | \([4]\) | \(4096\) | \(0.81261\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6840.g have rank \(0\).
Complex multiplication
The elliptic curves in class 6840.g do not have complex multiplication.Modular form 6840.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.