Show commands:
SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 407330.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
407330.g1 | 407330g3 | \([1, -1, 0, -11060960, 14161920240]\) | \(1010962818911303721/57392720\) | \(8496182327328080\) | \([2]\) | \(12615680\) | \(2.5231\) | \(\Gamma_0(N)\)-optimal* |
407330.g2 | 407330g4 | \([1, -1, 0, -1158080, -112835824]\) | \(1160306142246441/634128110000\) | \(93873718503739790000\) | \([2]\) | \(12615680\) | \(2.5231\) | |
407330.g3 | 407330g2 | \([1, -1, 0, -692560, 220569600]\) | \(248158561089321/1859334400\) | \(275248220852281600\) | \([2, 2]\) | \(6307840\) | \(2.1766\) | \(\Gamma_0(N)\)-optimal* |
407330.g4 | 407330g1 | \([1, -1, 0, -15440, 7818496]\) | \(-2749884201/176619520\) | \(-26146027657953280\) | \([2]\) | \(3153920\) | \(1.8300\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 407330.g have rank \(1\).
Complex multiplication
The elliptic curves in class 407330.g do not have complex multiplication.Modular form 407330.2.a.g
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.