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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 111090.bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 111090.bp1 | 111090bn4 | \([1, 1, 1, -19530732911, -1050542989215211]\) | \(5565604209893236690185614401/229307220930246900000\) | \(33945698304528506830994100000\) | \([2]\) | \(324403200\) | \(4.5559\) | |
| 111090.bp2 | 111090bn3 | \([1, 1, 1, -5954307631, 163082499823253]\) | \(157706830105239346386477121/13650704956054687500000\) | \(2020794243646261596679687500000\) | \([2]\) | \(324403200\) | \(4.5559\) | |
| 111090.bp3 | 111090bn2 | \([1, 1, 1, -1280232911, -14724911415211]\) | \(1567558142704512417614401/274462175610000000000\) | \(40630252163300467290000000000\) | \([2, 2]\) | \(162201600\) | \(4.2094\) | |
| 111090.bp4 | 111090bn1 | \([1, 1, 1, 152553009, -1318047004587]\) | \(2652277923951208297919/6605028468326400000\) | \(-977781261179006966169600000\) | \([2]\) | \(81100800\) | \(3.8628\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 111090.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 111090.bp do not have complex multiplication.Modular form 111090.2.a.bp
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
