Show commands:
SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 406560.bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
406560.bp1 | 406560bp2 | \([0, -1, 0, -38420081, 91674005601]\) | \(864335783029582144/59535\) | \(432004645416960\) | \([2]\) | \(19660800\) | \(2.7092\) | \(\Gamma_0(N)\)-optimal* |
406560.bp2 | 406560bp4 | \([0, -1, 0, -2697856, 1057037236]\) | \(2394165105226952/854262178245\) | \(774849310081991907840\) | \([2]\) | \(19660800\) | \(2.7092\) | |
406560.bp3 | 406560bp1 | \([0, -1, 0, -2401406, 1432817256]\) | \(13507798771700416/3544416225\) | \(401865571326542400\) | \([2, 2]\) | \(9830400\) | \(2.3626\) | \(\Gamma_0(N)\)-optimal* |
406560.bp4 | 406560bp3 | \([0, -1, 0, -2107376, 1796473560]\) | \(-1141100604753992/875529151875\) | \(-794139289510311360000\) | \([2]\) | \(19660800\) | \(2.7092\) |
Rank
sage: E.rank()
The elliptic curves in class 406560.bp have rank \(0\).
Complex multiplication
The elliptic curves in class 406560.bp do not have complex multiplication.Modular form 406560.2.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.