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SageMath
E = EllipticCurve("ko1")
E.isogeny_class()
Elliptic curves in class 418950ko
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 418950.ko4 | 418950ko1 | \([1, -1, 1, -111407855, 452635014647]\) | \(114113060120923921/124104960\) | \(166312515875940000000\) | \([2]\) | \(70778880\) | \(3.1674\) | \(\Gamma_0(N)\)-optimal |
| 418950.ko3 | 418950ko2 | \([1, -1, 1, -112289855, 445104498647]\) | \(116844823575501841/3760263939600\) | \(5039113313057348306250000\) | \([2, 2]\) | \(141557760\) | \(3.5140\) | |
| 418950.ko5 | 418950ko3 | \([1, -1, 1, 34342645, 1524612963647]\) | \(3342636501165359/751262567039460\) | \(-1006763691054327158354062500\) | \([2]\) | \(283115520\) | \(3.8606\) | |
| 418950.ko2 | 418950ko4 | \([1, -1, 1, -273034355, -1116367574353]\) | \(1679731262160129361/570261564022500\) | \(764205035962546589414062500\) | \([2, 2]\) | \(283115520\) | \(3.8606\) | |
| 418950.ko6 | 418950ko5 | \([1, -1, 1, 801572395, -7738094367853]\) | \(42502666283088696719/43898058864843750\) | \(-58827597316676745446777343750\) | \([2]\) | \(566231040\) | \(4.2072\) | |
| 418950.ko1 | 418950ko6 | \([1, -1, 1, -3919553105, -94430782386853]\) | \(4969327007303723277361/1123462695162150\) | \(1505547460191610492502343750\) | \([2]\) | \(566231040\) | \(4.2072\) |
Rank
sage: E.rank()
The elliptic curves in class 418950ko have rank \(1\).
Complex multiplication
The elliptic curves in class 418950ko do not have complex multiplication.Modular form 418950.2.a.ko
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
