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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 3192.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3192.b1 | 3192c3 | \([0, -1, 0, -592824, 175883580]\) | \(22501000029889239268/3620708343\) | \(3707605343232\) | \([4]\) | \(24576\) | \(1.8142\) | |
| 3192.b2 | 3192c2 | \([0, -1, 0, -37164, 2739924]\) | \(22174957026242512/278654127129\) | \(71335456545024\) | \([2, 2]\) | \(12288\) | \(1.4676\) | |
| 3192.b3 | 3192c4 | \([0, -1, 0, -6384, 7110684]\) | \(-28104147578308/21301741002339\) | \(-21812982786395136\) | \([2]\) | \(24576\) | \(1.8142\) | |
| 3192.b4 | 3192c1 | \([0, -1, 0, -4359, -41940]\) | \(572616640141312/280535480757\) | \(4488567692112\) | \([2]\) | \(6144\) | \(1.1210\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3192.b have rank \(0\).
Complex multiplication
The elliptic curves in class 3192.b do not have complex multiplication.Modular form 3192.2.a.b
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.
