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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 122694.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
122694.g1 | 122694x4 | \([1, 1, 0, -424102461, -3361840382259]\) | \(986551739719628473/111045168\) | \(949545741214032451632\) | \([2]\) | \(34406400\) | \(3.4495\) | |
122694.g2 | 122694x3 | \([1, 1, 0, -47840861, 43177247469]\) | \(1416134368422073/725251155408\) | \(6201612896188538344765392\) | \([2]\) | \(34406400\) | \(3.4495\) | |
122694.g3 | 122694x2 | \([1, 1, 0, -26573901, -52256108835]\) | \(242702053576633/2554695936\) | \(21845170661776083857664\) | \([2, 2]\) | \(17203200\) | \(3.1029\) | |
122694.g4 | 122694x1 | \([1, 1, 0, -399181, -2026821155]\) | \(-822656953/207028224\) | \(-1770295564866944434176\) | \([2]\) | \(8601600\) | \(2.7564\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 122694.g have rank \(0\).
Complex multiplication
The elliptic curves in class 122694.g do not have complex multiplication.Modular form 122694.2.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.