Show commands:
SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 127680j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 127680.d4 | 127680j1 | \([0, -1, 0, -646721, 200396865]\) | \(114113060120923921/124104960\) | \(32533370634240\) | \([2]\) | \(1474560\) | \(1.8802\) | \(\Gamma_0(N)\)-optimal |
| 127680.d3 | 127680j2 | \([0, -1, 0, -651841, 197067841]\) | \(116844823575501841/3760263939600\) | \(985730630182502400\) | \([2, 2]\) | \(2949120\) | \(2.2268\) | |
| 127680.d5 | 127680j3 | \([0, -1, 0, 199359, 674250561]\) | \(3342636501165359/751262567039460\) | \(-196938974373992202240\) | \([2]\) | \(5898240\) | \(2.5733\) | |
| 127680.d2 | 127680j4 | \([0, -1, 0, -1584961, -493254335]\) | \(1679731262160129361/570261564022500\) | \(149490647439114240000\) | \([2, 2]\) | \(5898240\) | \(2.5733\) | |
| 127680.d6 | 127680j5 | \([0, -1, 0, 4653119, -3423904319]\) | \(42502666283088696719/43898058864843750\) | \(-11507612743065600000000\) | \([2]\) | \(11796480\) | \(2.9199\) | |
| 127680.d1 | 127680j6 | \([0, -1, 0, -22752961, -41758153535]\) | \(4969327007303723277361/1123462695162150\) | \(294509004760586649600\) | \([2]\) | \(11796480\) | \(2.9199\) |
Rank
sage: E.rank()
The elliptic curves in class 127680j have rank \(0\).
Complex multiplication
The elliptic curves in class 127680j do not have complex multiplication.Modular form 127680.2.a.j
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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
