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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 510.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
510.g1 | 510g4 | \([1, 0, 0, -52405, -4621873]\) | \(15916310615119911121/2210850\) | \(2210850\) | \([2]\) | \(864\) | \(1.0716\) | |
510.g2 | 510g3 | \([1, 0, 0, -3275, -72435]\) | \(-3884775383991601/1448254140\) | \(-1448254140\) | \([2]\) | \(432\) | \(0.72501\) | |
510.g3 | 510g2 | \([1, 0, 0, -655, -6223]\) | \(31080575499121/1549125000\) | \(1549125000\) | \([6]\) | \(288\) | \(0.52227\) | |
510.g4 | 510g1 | \([1, 0, 0, 25, -375]\) | \(1723683599/62424000\) | \(-62424000\) | \([6]\) | \(144\) | \(0.17570\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 510.g have rank \(0\).
Complex multiplication
The elliptic curves in class 510.g do not have complex multiplication.Modular form 510.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.