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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 3192.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3192.i1 | 3192a3 | \([0, -1, 0, -1232, 17052]\) | \(202119559492/136857\) | \(140141568\) | \([2]\) | \(2048\) | \(0.50119\) | |
3192.i2 | 3192a2 | \([0, -1, 0, -92, 180]\) | \(340062928/159201\) | \(40755456\) | \([2, 2]\) | \(1024\) | \(0.15462\) | |
3192.i3 | 3192a1 | \([0, -1, 0, -47, -108]\) | \(733001728/10773\) | \(172368\) | \([2]\) | \(512\) | \(-0.19195\) | \(\Gamma_0(N)\)-optimal |
3192.i4 | 3192a4 | \([0, -1, 0, 328, 1020]\) | \(3799448348/2736741\) | \(-2802422784\) | \([2]\) | \(2048\) | \(0.50119\) |
Rank
sage: E.rank()
The elliptic curves in class 3192.i have rank \(0\).
Complex multiplication
The elliptic curves in class 3192.i do not have complex multiplication.Modular form 3192.2.a.i
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.