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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 3380.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3380.c1 | 3380i3 | \([0, 1, 0, -6985, -226992]\) | \(488095744/125\) | \(9653618000\) | \([2]\) | \(3456\) | \(0.90183\) | |
3380.c2 | 3380i4 | \([0, 1, 0, -6140, -283100]\) | \(-20720464/15625\) | \(-19307236000000\) | \([2]\) | \(6912\) | \(1.2484\) | |
3380.c3 | 3380i1 | \([0, 1, 0, -225, 820]\) | \(16384/5\) | \(386144720\) | \([2]\) | \(1152\) | \(0.35252\) | \(\Gamma_0(N)\)-optimal |
3380.c4 | 3380i2 | \([0, 1, 0, 620, 6228]\) | \(21296/25\) | \(-30891577600\) | \([2]\) | \(2304\) | \(0.69910\) |
Rank
sage: E.rank()
The elliptic curves in class 3380.c have rank \(1\).
Complex multiplication
The elliptic curves in class 3380.c do not have complex multiplication.Modular form 3380.2.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.