Show commands:
SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 199962.bk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
199962.bk1 | 199962k3 | \([1, -1, 1, -5057075, 4378472173]\) | \(-545407363875/14\) | \(-367137590801562\) | \([]\) | \(3528360\) | \(2.3109\) | |
199962.bk2 | 199962k2 | \([1, -1, 1, -58025, 6902929]\) | \(-7414875/2744\) | \(-7995440866345128\) | \([]\) | \(1176120\) | \(1.7616\) | |
199962.bk3 | 199962k1 | \([1, -1, 1, 5455, -96799]\) | \(4492125/3584\) | \(-14325136906752\) | \([]\) | \(392040\) | \(1.2123\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 199962.bk have rank \(0\).
Complex multiplication
The elliptic curves in class 199962.bk do not have complex multiplication.Modular form 199962.2.a.bk
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.