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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 45080bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 45080.r4 | 45080bg1 | \([0, 0, 0, 1078, -31899]\) | \(73598976/276115\) | \(-519754458160\) | \([2]\) | \(30720\) | \(0.93088\) | \(\Gamma_0(N)\)-optimal |
| 45080.r3 | 45080bg2 | \([0, 0, 0, -10927, -384846]\) | \(4790692944/648025\) | \(19517310265600\) | \([2, 2]\) | \(61440\) | \(1.2775\) | |
| 45080.r2 | 45080bg3 | \([0, 0, 0, -45227, 3312694]\) | \(84923690436/9794435\) | \(1179960814914560\) | \([4]\) | \(122880\) | \(1.6240\) | |
| 45080.r1 | 45080bg4 | \([0, 0, 0, -168707, -26670994]\) | \(4407931365156/100625\) | \(12122552960000\) | \([2]\) | \(122880\) | \(1.6240\) |
Rank
sage: E.rank()
The elliptic curves in class 45080bg have rank \(1\).
Complex multiplication
The elliptic curves in class 45080bg do not have complex multiplication.Modular form 45080.2.a.bg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
