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Properties

Label 10005.b
Number of curves $2$
Conductor $10005$
CM no
Rank $1$
Graph

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

Elliptic curves in class 10005.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10005.b1 10005h2 \([1, 1, 1, -1380, 19152]\) \(290656902035521/86293125\) \(86293125\) \([2]\) \(5888\) \(0.50129\)  
10005.b2 10005h1 \([1, 1, 1, -75, 360]\) \(-46694890801/39169575\) \(-39169575\) \([2]\) \(2944\) \(0.15472\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 10005.b have rank \(1\).

Complex multiplication

The elliptic curves in class 10005.b do not have complex multiplication.

Modular form 10005.2.a.b

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