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SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 10320v
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10320.k4 | 10320v1 | \([0, -1, 0, -1720, -7568]\) | \(137467988281/72562500\) | \(297216000000\) | \([2]\) | \(11520\) | \(0.89304\) | \(\Gamma_0(N)\)-optimal |
10320.k3 | 10320v2 | \([0, -1, 0, -21720, -1223568]\) | \(276670733768281/336980250\) | \(1380271104000\) | \([2]\) | \(23040\) | \(1.2396\) | |
10320.k2 | 10320v3 | \([0, -1, 0, -79720, 8690032]\) | \(13679527032530281/381633600\) | \(1563171225600\) | \([2]\) | \(34560\) | \(1.4423\) | |
10320.k1 | 10320v4 | \([0, -1, 0, -82920, 7957872]\) | \(15393836938735081/2275690697640\) | \(9321229097533440\) | \([2]\) | \(69120\) | \(1.7889\) |
Rank
sage: E.rank()
The elliptic curves in class 10320v have rank \(1\).
Complex multiplication
The elliptic curves in class 10320v do not have complex multiplication.Modular form 10320.2.a.v
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 Cremona 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 Cremona labels.