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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 180336bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 180336.k4 | 180336bc1 | \([0, -1, 0, -90264, -217890576]\) | \(-822656953/207028224\) | \(-20468359338997579776\) | \([2]\) | \(4915200\) | \(2.3847\) | \(\Gamma_0(N)\)-optimal |
| 180336.k3 | 180336bc2 | \([0, -1, 0, -6008984, -5615763216]\) | \(242702053576633/2554695936\) | \(252576356062083416064\) | \([2, 2]\) | \(9830400\) | \(2.7313\) | |
| 180336.k2 | 180336bc3 | \([0, -1, 0, -10817944, 4648481008]\) | \(1416134368422073/725251155408\) | \(71703756005336363630592\) | \([2]\) | \(19660800\) | \(3.0778\) | |
| 180336.k1 | 180336bc4 | \([0, -1, 0, -95899544, -361438555920]\) | \(986551739719628473/111045168\) | \(10978756217719160832\) | \([2]\) | \(19660800\) | \(3.0778\) |
Rank
sage: E.rank()
The elliptic curves in class 180336bc have rank \(1\).
Complex multiplication
The elliptic curves in class 180336bc do not have complex multiplication.Modular form 180336.2.a.bc
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.
