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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 5070.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5070.k1 | 5070k4 | \([1, 0, 1, -202128, -33214244]\) | \(189208196468929/10860320250\) | \(52420691525582250\) | \([2]\) | \(48384\) | \(1.9623\) | |
| 5070.k2 | 5070k2 | \([1, 0, 1, -34818, 2486668]\) | \(967068262369/4928040\) | \(23786707824360\) | \([2]\) | \(16128\) | \(1.4130\) | |
| 5070.k3 | 5070k1 | \([1, 0, 1, -1018, 80108]\) | \(-24137569/561600\) | \(-2710735934400\) | \([2]\) | \(8064\) | \(1.0664\) | \(\Gamma_0(N)\)-optimal |
| 5070.k4 | 5070k3 | \([1, 0, 1, 9122, -2118244]\) | \(17394111071/411937500\) | \(-1988343632437500\) | \([2]\) | \(24192\) | \(1.6157\) |
Rank
sage: E.rank()
The elliptic curves in class 5070.k have rank \(1\).
Complex multiplication
The elliptic curves in class 5070.k do not have complex multiplication.Modular form 5070.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
