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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 12870.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12870.k1 | 12870o3 | \([1, -1, 0, -196515074745, -33530596817126675]\) | \(1151287518770166280399859009187288721/877598977782384000\) | \(639769654803357936000\) | \([2]\) | \(37158912\) | \(4.6844\) | |
| 12870.k2 | 12870o4 | \([1, -1, 0, -12369314745, -516103378502675]\) | \(287099942490903701230558394328721/8299347173197257908489616000\) | \(6050224089260801015288930064000\) | \([2]\) | \(37158912\) | \(4.6844\) | |
| 12870.k3 | 12870o2 | \([1, -1, 0, -12282194745, -523913041814675]\) | \(281076231077501634961715630808721/245403072288481536000000\) | \(178898839698303039744000000\) | \([2, 2]\) | \(18579456\) | \(4.3378\) | |
| 12870.k4 | 12870o1 | \([1, -1, 0, -762194745, -8307793814675]\) | \(-67172890180943415009710808721/2029083623424000000000000\) | \(-1479201961476096000000000000\) | \([2]\) | \(9289728\) | \(3.9912\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12870.k have rank \(0\).
Complex multiplication
The elliptic curves in class 12870.k do not have complex multiplication.Modular form 12870.2.a.k
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
