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Properties

Label 18735.b
Number of curves $2$
Conductor $18735$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 18735.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
18735.b1 18735b2 \([1, 0, 0, -106, 341]\) \(131794519969/23400015\) \(23400015\) \([2]\) \(5632\) \(0.13345\)  
18735.b2 18735b1 \([1, 0, 0, -31, -64]\) \(3301293169/281025\) \(281025\) \([2]\) \(2816\) \(-0.21312\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 18735.b have rank \(1\).

Complex multiplication

The elliptic curves in class 18735.b do not have complex multiplication.

Modular form 18735.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} + 4 q^{7} + 3 q^{8} + q^{9} + q^{10} + 4 q^{11} - q^{12} - 2 q^{13} - 4 q^{14} - q^{15} - q^{16} - 6 q^{17} - q^{18} + 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.