Properties

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

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 90354a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
90354.c4 90354a1 \([1, 1, 0, -2766, -156]\) \(912673/528\) \(1354703543952\) \([2]\) \(202752\) \(1.0174\) \(\Gamma_0(N)\)-optimal
90354.c2 90354a2 \([1, 1, 0, -30146, -2020800]\) \(1180932193/4356\) \(11176304237604\) \([2, 2]\) \(405504\) \(1.3639\)  
90354.c3 90354a3 \([1, 1, 0, -16456, -3847046]\) \(-192100033/2371842\) \(-6085497657375378\) \([2]\) \(811008\) \(1.7105\)  
90354.c1 90354a4 \([1, 1, 0, -481916, -128968170]\) \(4824238966273/66\) \(169337942994\) \([2]\) \(811008\) \(1.7105\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 90354.2.a.a

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.