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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 11970r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 11970.c4 | 11970r1 | \([1, -1, 0, -90945, -10533699]\) | \(114113060120923921/124104960\) | \(90472515840\) | \([2]\) | \(61440\) | \(1.3898\) | \(\Gamma_0(N)\)-optimal |
| 11970.c3 | 11970r2 | \([1, -1, 0, -91665, -10357875]\) | \(116844823575501841/3760263939600\) | \(2741232411968400\) | \([2, 2]\) | \(122880\) | \(1.7363\) | |
| 11970.c2 | 11970r3 | \([1, -1, 0, -222885, 26095041]\) | \(1679731262160129361/570261564022500\) | \(415720680172402500\) | \([2, 2]\) | \(245760\) | \(2.0829\) | |
| 11970.c5 | 11970r4 | \([1, -1, 0, 28035, -35566695]\) | \(3342636501165359/751262567039460\) | \(-547670411371766340\) | \([2]\) | \(245760\) | \(2.0829\) | |
| 11970.c1 | 11970r5 | \([1, -1, 0, -3199635, 2203289991]\) | \(4969327007303723277361/1123462695162150\) | \(819004304773207350\) | \([2]\) | \(491520\) | \(2.4295\) | |
| 11970.c6 | 11970r6 | \([1, -1, 0, 654345, 180312075]\) | \(42502666283088696719/43898058864843750\) | \(-32001684912471093750\) | \([2]\) | \(491520\) | \(2.4295\) |
Rank
sage: E.rank()
The elliptic curves in class 11970r have rank \(0\).
Complex multiplication
The elliptic curves in class 11970r do not have complex multiplication.Modular form 11970.2.a.r
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
