Show commands:
SageMath
E = EllipticCurve("ms1")
E.isogeny_class()
Elliptic curves in class 235200.ms
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
235200.ms1 | 235200ms6 | \([0, -1, 0, -1317121633, -18398252484863]\) | \(524388516989299201/3150\) | \(1517954457600000000\) | \([2]\) | \(56623104\) | \(3.5542\) | |
235200.ms2 | 235200ms4 | \([0, -1, 0, -82321633, -287440884863]\) | \(128031684631201/9922500\) | \(4781556541440000000000\) | \([2, 2]\) | \(28311552\) | \(3.2076\) | |
235200.ms3 | 235200ms5 | \([0, -1, 0, -76833633, -327420964863]\) | \(-104094944089921/35880468750\) | \(-17290449993600000000000000\) | \([2]\) | \(56623104\) | \(3.5542\) | |
235200.ms4 | 235200ms3 | \([0, -1, 0, -29009633, 56851147137]\) | \(5602762882081/345888060\) | \(166680102383370240000000\) | \([2]\) | \(28311552\) | \(3.2076\) | |
235200.ms5 | 235200ms2 | \([0, -1, 0, -5489633, -3853972863]\) | \(37966934881/8643600\) | \(4165267031654400000000\) | \([2, 2]\) | \(14155776\) | \(2.8610\) | |
235200.ms6 | 235200ms1 | \([0, -1, 0, 782367, -373012863]\) | \(109902239/188160\) | \(-90672479600640000000\) | \([2]\) | \(7077888\) | \(2.5144\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 235200.ms have rank \(1\).
Complex multiplication
The elliptic curves in class 235200.ms do not have complex multiplication.Modular form 235200.2.a.ms
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.