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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 82800.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
82800.g1 | 82800bc4 | \([0, 0, 0, -169275, 22684250]\) | \(45989074372/7555707\) | \(88129766448000000\) | \([2]\) | \(786432\) | \(1.9731\) | |
82800.g2 | 82800bc2 | \([0, 0, 0, -47775, -3681250]\) | \(4135597648/385641\) | \(1124529156000000\) | \([2, 2]\) | \(393216\) | \(1.6265\) | |
82800.g3 | 82800bc1 | \([0, 0, 0, -46650, -3878125]\) | \(61604313088/621\) | \(113177250000\) | \([2]\) | \(196608\) | \(1.2799\) | \(\Gamma_0(N)\)-optimal |
82800.g4 | 82800bc3 | \([0, 0, 0, 55725, -17446750]\) | \(1640689628/12223143\) | \(-142570739952000000\) | \([2]\) | \(786432\) | \(1.9731\) |
Rank
sage: E.rank()
The elliptic curves in class 82800.g have rank \(0\).
Complex multiplication
The elliptic curves in class 82800.g do not have complex multiplication.Modular form 82800.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.