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Properties

Label 90354.k
Number of curves $4$
Conductor $90354$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 90354.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
90354.k1 90354k3 \([1, 0, 1, -13779014, -19687955836]\) \(112763292123580561/1932612\) \(4958553646750308\) \([2]\) \(5184000\) \(2.5533\)  
90354.k2 90354k4 \([1, 0, 1, -13765324, -19729025836]\) \(-112427521449300721/466873642818\) \(-1197870035044175780562\) \([2]\) \(10368000\) \(2.8999\)  
90354.k3 90354k1 \([1, 0, 1, -61634, 4287764]\) \(10091699281/2737152\) \(7022783171847168\) \([2]\) \(1036800\) \(1.7486\) \(\Gamma_0(N)\)-optimal
90354.k4 90354k2 \([1, 0, 1, 157406, 27944084]\) \(168105213359/228637728\) \(-586621856823358752\) \([2]\) \(2073600\) \(2.0952\)  

Rank

sage: E.rank()
 

The elliptic curves in class 90354.k have rank \(1\).

Complex multiplication

The elliptic curves in class 90354.k do not have complex multiplication.

Modular form 90354.2.a.k

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

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.