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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 912.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
912.c1 | 912e3 | \([0, -1, 0, -6848, 220416]\) | \(8671983378625/82308\) | \(337133568\) | \([2]\) | \(864\) | \(0.79865\) | |
912.c2 | 912e4 | \([0, -1, 0, -6688, 231040]\) | \(-8078253774625/846825858\) | \(-3468598714368\) | \([2]\) | \(1728\) | \(1.1452\) | |
912.c3 | 912e1 | \([0, -1, 0, -128, 0]\) | \(57066625/32832\) | \(134479872\) | \([2]\) | \(288\) | \(0.24934\) | \(\Gamma_0(N)\)-optimal |
912.c4 | 912e2 | \([0, -1, 0, 512, -512]\) | \(3616805375/2105352\) | \(-8623521792\) | \([2]\) | \(576\) | \(0.59592\) |
Rank
sage: E.rank()
The elliptic curves in class 912.c have rank \(0\).
Complex multiplication
The elliptic curves in class 912.c do not have complex multiplication.Modular form 912.2.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.