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SageMath
E = EllipticCurve("fd1")
E.isogeny_class()
Elliptic curves in class 61200fd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 61200.w4 | 61200fd1 | \([0, 0, 0, -10875, 270250]\) | \(3048625/1088\) | \(50761728000000\) | \([2]\) | \(165888\) | \(1.3308\) | \(\Gamma_0(N)\)-optimal |
| 61200.w3 | 61200fd2 | \([0, 0, 0, -154875, 23454250]\) | \(8805624625/2312\) | \(107868672000000\) | \([2]\) | \(331776\) | \(1.6774\) | |
| 61200.w2 | 61200fd3 | \([0, 0, 0, -370875, -86921750]\) | \(120920208625/19652\) | \(916883712000000\) | \([2]\) | \(497664\) | \(1.8801\) | |
| 61200.w1 | 61200fd4 | \([0, 0, 0, -406875, -69029750]\) | \(159661140625/48275138\) | \(2252324838528000000\) | \([2]\) | \(995328\) | \(2.2267\) |
Rank
sage: E.rank()
The elliptic curves in class 61200fd have rank \(1\).
Complex multiplication
The elliptic curves in class 61200fd do not have complex multiplication.Modular form 61200.2.a.fd
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.
