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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 1600.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1600.c1 | 1600g3 | \([0, 1, 0, -4133, -103637]\) | \(488095744/125\) | \(2000000000\) | \([2]\) | \(1152\) | \(0.77065\) | |
1600.c2 | 1600g4 | \([0, 1, 0, -3633, -129137]\) | \(-20720464/15625\) | \(-4000000000000\) | \([2]\) | \(2304\) | \(1.1172\) | |
1600.c3 | 1600g1 | \([0, 1, 0, -133, 363]\) | \(16384/5\) | \(80000000\) | \([2]\) | \(384\) | \(0.22134\) | \(\Gamma_0(N)\)-optimal |
1600.c4 | 1600g2 | \([0, 1, 0, 367, 2863]\) | \(21296/25\) | \(-6400000000\) | \([2]\) | \(768\) | \(0.56792\) |
Rank
sage: E.rank()
The elliptic curves in class 1600.c have rank \(1\).
Complex multiplication
The elliptic curves in class 1600.c do not have complex multiplication.Modular form 1600.2.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.