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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 12342.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12342.j1 | 12342q1 | \([1, 0, 1, -109387, -13824010]\) | \(81706955619457/744505344\) | \(1318936631721984\) | \([2]\) | \(134400\) | \(1.7235\) | \(\Gamma_0(N)\)-optimal |
12342.j2 | 12342q2 | \([1, 0, 1, -31947, -32998154]\) | \(-2035346265217/264305213568\) | \(-468232808453739648\) | \([2]\) | \(268800\) | \(2.0701\) |
Rank
sage: E.rank()
The elliptic curves in class 12342.j have rank \(0\).
Complex multiplication
The elliptic curves in class 12342.j do not have complex multiplication.Modular form 12342.2.a.j
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.