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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 342.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
342.c1 | 342c3 | \([1, -1, 0, -3852, 92988]\) | \(8671983378625/82308\) | \(60002532\) | \([6]\) | \(288\) | \(0.65481\) | |
342.c2 | 342c4 | \([1, -1, 0, -3762, 97470]\) | \(-8078253774625/846825858\) | \(-617336050482\) | \([6]\) | \(576\) | \(1.0014\) | |
342.c3 | 342c1 | \([1, -1, 0, -72, 0]\) | \(57066625/32832\) | \(23934528\) | \([2]\) | \(96\) | \(0.10550\) | \(\Gamma_0(N)\)-optimal |
342.c4 | 342c2 | \([1, -1, 0, 288, -216]\) | \(3616805375/2105352\) | \(-1534801608\) | \([2]\) | \(192\) | \(0.45208\) |
Rank
sage: E.rank()
The elliptic curves in class 342.c have rank \(1\).
Complex multiplication
The elliptic curves in class 342.c do not have complex multiplication.Modular form 342.2.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.