Show commands:
SageMath
E = EllipticCurve("ch1")
E.isogeny_class()
Elliptic curves in class 45570.ch
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 45570.ch1 | 45570cb4 | \([1, 1, 1, -31138080, 66865420677]\) | \(28379906689597370652529/1357352437500\) | \(159691156919437500\) | \([2]\) | \(3110400\) | \(2.7783\) | |
| 45570.ch2 | 45570cb3 | \([1, 1, 1, -1942900, 1047806885]\) | \(-6894246873502147249/47925198774000\) | \(-5638351710562326000\) | \([2]\) | \(1555200\) | \(2.4317\) | |
| 45570.ch3 | 45570cb2 | \([1, 1, 1, -418020, 74580645]\) | \(68663623745397169/19216056254400\) | \(2260749802273905600\) | \([2]\) | \(1036800\) | \(2.2290\) | |
| 45570.ch4 | 45570cb1 | \([1, 1, 1, 68060, 7696037]\) | \(296354077829711/387386634240\) | \(-45575650131701760\) | \([2]\) | \(518400\) | \(1.8824\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 45570.ch have rank \(0\).
Complex multiplication
The elliptic curves in class 45570.ch do not have complex multiplication.Modular form 45570.2.a.ch
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
