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SageMath
E = EllipticCurve("dl1")
E.isogeny_class()
Elliptic curves in class 172480dl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 172480.gy4 | 172480dl1 | \([0, -1, 0, 31295, 17405025]\) | \(109902239/4312000\) | \(-132986303414272000\) | \([2]\) | \(1769472\) | \(1.9656\) | \(\Gamma_0(N)\)-optimal |
| 172480.gy2 | 172480dl2 | \([0, -1, 0, -846785, 287326817]\) | \(2177286259681/105875000\) | \(3265288699904000000\) | \([2]\) | \(3538944\) | \(2.3122\) | |
| 172480.gy3 | 172480dl3 | \([0, -1, 0, -282305, -476264095]\) | \(-80677568161/3131816380\) | \(-96588284635544289280\) | \([2]\) | \(5308416\) | \(2.5149\) | |
| 172480.gy1 | 172480dl4 | \([0, -1, 0, -11038785, -14035882783]\) | \(4823468134087681/30382271150\) | \(937019000363771494400\) | \([2]\) | \(10616832\) | \(2.8615\) |
Rank
sage: E.rank()
The elliptic curves in class 172480dl have rank \(0\).
Complex multiplication
The elliptic curves in class 172480dl do not have complex multiplication.Modular form 172480.2.a.dl
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
