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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 12870c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12870.h3 | 12870c1 | \([1, -1, 0, -7215, 668781]\) | \(-1538518817843307/6227391227200\) | \(-168139563134400\) | \([6]\) | \(55296\) | \(1.4149\) | \(\Gamma_0(N)\)-optimal |
| 12870.h2 | 12870c2 | \([1, -1, 0, -166935, 26255925]\) | \(19054765821218746347/32122413895000\) | \(867305175165000\) | \([6]\) | \(110592\) | \(1.7615\) | |
| 12870.h4 | 12870c3 | \([1, -1, 0, 63570, -16000300]\) | \(1443395048293197/6443008000000\) | \(-126817726464000000\) | \([2]\) | \(165888\) | \(1.9643\) | |
| 12870.h1 | 12870c4 | \([1, -1, 0, -696750, -198629164]\) | \(1900481745258486963/232375000000000\) | \(4573837125000000000\) | \([2]\) | \(331776\) | \(2.3108\) |
Rank
sage: E.rank()
The elliptic curves in class 12870c have rank \(1\).
Complex multiplication
The elliptic curves in class 12870c do not have complex multiplication.Modular form 12870.2.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 Cremona 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 Cremona labels.
