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SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 12870.cc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12870.cc1 | 12870bj4 | \([1, -1, 1, -1502417, -707407559]\) | \(19054765821218746347/32122413895000\) | \(632265472695285000\) | \([2]\) | \(331776\) | \(2.3108\) | |
| 12870.cc2 | 12870bj2 | \([1, -1, 1, -77417, 7382441]\) | \(1900481745258486963/232375000000000\) | \(6274125000000000\) | \([6]\) | \(110592\) | \(1.7615\) | |
| 12870.cc3 | 12870bj3 | \([1, -1, 1, -64937, -17992151]\) | \(-1538518817843307/6227391227200\) | \(-122573741524977600\) | \([2]\) | \(165888\) | \(1.9643\) | |
| 12870.cc4 | 12870bj1 | \([1, -1, 1, 7063, 590249]\) | \(1443395048293197/6443008000000\) | \(-173961216000000\) | \([6]\) | \(55296\) | \(1.4149\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12870.cc have rank \(0\).
Complex multiplication
The elliptic curves in class 12870.cc do not have complex multiplication.Modular form 12870.2.a.cc
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
