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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 267696bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 267696.bp4 | 267696bp1 | \([0, 0, 0, 1014, 340535]\) | \(2048/891\) | \(-50163211056816\) | \([2]\) | \(737280\) | \(1.3080\) | \(\Gamma_0(N)\)-optimal |
| 267696.bp3 | 267696bp2 | \([0, 0, 0, -67431, 6569030]\) | \(37642192/1089\) | \(980969460666624\) | \([2, 2]\) | \(1474560\) | \(1.6545\) | |
| 267696.bp1 | 267696bp3 | \([0, 0, 0, -1071291, 426784826]\) | \(37736227588/33\) | \(118905389171712\) | \([2]\) | \(2949120\) | \(2.0011\) | |
| 267696.bp2 | 267696bp4 | \([0, 0, 0, -158691, -15023086]\) | \(122657188/43923\) | \(158263072987548672\) | \([2]\) | \(2949120\) | \(2.0011\) |
Rank
sage: E.rank()
The elliptic curves in class 267696bp have rank \(1\).
Complex multiplication
The elliptic curves in class 267696bp do not have complex multiplication.Modular form 267696.2.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 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.
