Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1859.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1859.b1 | 1859a3 | \([0, -1, 1, -1321636, -584371175]\) | \(-52893159101157376/11\) | \(-53094899\) | \([]\) | \(10800\) | \(1.7792\) | |
1859.b2 | 1859a2 | \([0, -1, 1, -1746, -50295]\) | \(-122023936/161051\) | \(-777362416259\) | \([]\) | \(2160\) | \(0.97446\) | |
1859.b3 | 1859a1 | \([0, -1, 1, -56, 405]\) | \(-4096/11\) | \(-53094899\) | \([]\) | \(432\) | \(0.16975\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1859.b have rank \(1\).
Complex multiplication
The elliptic curves in class 1859.b do not have complex multiplication.Modular form 1859.2.a.b
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.
\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.