Properties

Label 1859.b
Number of curves $3$
Conductor $1859$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
sage: E = EllipticCurve("b1")
 
sage: 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)
 
\(q + 2 q^{2} - q^{3} + 2 q^{4} - q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{9} - 2 q^{10} - q^{11} - 2 q^{12} + 4 q^{14} + q^{15} - 4 q^{16} - 2 q^{17} - 4 q^{18} + O(q^{20})\) Copy content Toggle raw display

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.