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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 10580.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10580.c1 | 10580m3 | \([0, 1, 0, -21865, 1236900]\) | \(488095744/125\) | \(296071778000\) | \([2]\) | \(17820\) | \(1.1871\) | |
10580.c2 | 10580m4 | \([0, 1, 0, -19220, 1550068]\) | \(-20720464/15625\) | \(-592143556000000\) | \([2]\) | \(35640\) | \(1.5337\) | |
10580.c3 | 10580m1 | \([0, 1, 0, -705, -5192]\) | \(16384/5\) | \(11842871120\) | \([2]\) | \(5940\) | \(0.63780\) | \(\Gamma_0(N)\)-optimal |
10580.c4 | 10580m2 | \([0, 1, 0, 1940, -32700]\) | \(21296/25\) | \(-947429689600\) | \([2]\) | \(11880\) | \(0.98437\) |
Rank
sage: E.rank()
The elliptic curves in class 10580.c have rank \(0\).
Complex multiplication
The elliptic curves in class 10580.c do not have complex multiplication.Modular form 10580.2.a.c
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.