Show commands:
SageMath
E = EllipticCurve("gp1")
E.isogeny_class()
Elliptic curves in class 129600.gp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
129600.gp1 | 129600fy4 | \([0, 0, 0, -15511500, 23514138000]\) | \(-189613868625/128\) | \(-278628139008000000\) | \([]\) | \(3483648\) | \(2.6631\) | |
129600.gp2 | 129600fy3 | \([0, 0, 0, -151500, 46106000]\) | \(-1159088625/2097152\) | \(-695784701952000000\) | \([]\) | \(1161216\) | \(2.1138\) | |
129600.gp3 | 129600fy1 | \([0, 0, 0, -7500, -262000]\) | \(-140625/8\) | \(-2654208000000\) | \([]\) | \(165888\) | \(1.1409\) | \(\Gamma_0(N)\)-optimal |
129600.gp4 | 129600fy2 | \([0, 0, 0, 40500, -486000]\) | \(3375/2\) | \(-4353564672000000\) | \([]\) | \(497664\) | \(1.6902\) |
Rank
sage: E.rank()
The elliptic curves in class 129600.gp have rank \(0\).
Complex multiplication
The elliptic curves in class 129600.gp do not have complex multiplication.Modular form 129600.2.a.gp
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 21 & 7 \\ 3 & 1 & 7 & 21 \\ 21 & 7 & 1 & 3 \\ 7 & 21 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.