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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 13200.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
13200.c1 | 13200bu3 | \([0, -1, 0, -140808, 20384112]\) | \(4824238966273/66\) | \(4224000000\) | \([2]\) | \(49152\) | \(1.4029\) | |
13200.c2 | 13200bu2 | \([0, -1, 0, -8808, 320112]\) | \(1180932193/4356\) | \(278784000000\) | \([2, 2]\) | \(24576\) | \(1.0563\) | |
13200.c3 | 13200bu4 | \([0, -1, 0, -4808, 608112]\) | \(-192100033/2371842\) | \(-151797888000000\) | \([2]\) | \(49152\) | \(1.4029\) | |
13200.c4 | 13200bu1 | \([0, -1, 0, -808, 112]\) | \(912673/528\) | \(33792000000\) | \([2]\) | \(12288\) | \(0.70977\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 13200.c have rank \(1\).
Complex multiplication
The elliptic curves in class 13200.c do not have complex multiplication.Modular form 13200.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.