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SageMath
E = EllipticCurve("gi1")
E.isogeny_class()
Elliptic curves in class 232050gi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 232050.gi3 | 232050gi1 | \([1, 0, 0, -60463, 5717417]\) | \(1564491509212969/1856400\) | \(29006250000\) | \([2]\) | \(884736\) | \(1.2898\) | \(\Gamma_0(N)\)-optimal |
| 232050.gi2 | 232050gi2 | \([1, 0, 0, -60963, 5617917]\) | \(1603626125868649/53847202500\) | \(841362539062500\) | \([2, 2]\) | \(1769472\) | \(1.6364\) | |
| 232050.gi4 | 232050gi3 | \([1, 0, 0, 20287, 19511667]\) | \(59095693799351/10558110940650\) | \(-164970483447656250\) | \([2]\) | \(3538944\) | \(1.9830\) | |
| 232050.gi1 | 232050gi4 | \([1, 0, 0, -150213, -14641833]\) | \(23989788887201929/7965841406250\) | \(124466271972656250\) | \([2]\) | \(3538944\) | \(1.9830\) |
Rank
sage: E.rank()
The elliptic curves in class 232050gi have rank \(0\).
Complex multiplication
The elliptic curves in class 232050gi do not have complex multiplication.Modular form 232050.2.a.gi
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
