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SageMath
E = EllipticCurve("fr1")
E.isogeny_class()
Elliptic curves in class 58800.fr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 58800.fr1 | 58800iz6 | \([0, 1, 0, -329280408, -2299946200812]\) | \(524388516989299201/3150\) | \(23718038400000000\) | \([2]\) | \(7077888\) | \(3.2076\) | |
| 58800.fr2 | 58800iz4 | \([0, 1, 0, -20580408, -35940400812]\) | \(128031684631201/9922500\) | \(74711820960000000000\) | \([2, 2]\) | \(3538944\) | \(2.8610\) | |
| 58800.fr3 | 58800iz5 | \([0, 1, 0, -19208408, -40937224812]\) | \(-104094944089921/35880468750\) | \(-270163281150000000000000\) | \([2]\) | \(7077888\) | \(3.2076\) | |
| 58800.fr4 | 58800iz3 | \([0, 1, 0, -7252408, 7102767188]\) | \(5602762882081/345888060\) | \(2604376599740160000000\) | \([2]\) | \(3538944\) | \(2.8610\) | |
| 58800.fr5 | 58800iz2 | \([0, 1, 0, -1372408, -482432812]\) | \(37966934881/8643600\) | \(65082297369600000000\) | \([2, 2]\) | \(1769472\) | \(2.5144\) | |
| 58800.fr6 | 58800iz1 | \([0, 1, 0, 195592, -46528812]\) | \(109902239/188160\) | \(-1416757493760000000\) | \([2]\) | \(884736\) | \(2.1679\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 58800.fr have rank \(0\).
Complex multiplication
The elliptic curves in class 58800.fr do not have complex multiplication.Modular form 58800.2.a.fr
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
