Properties

Label 90.b
Number of curves $4$
Conductor $90$
CM no
Rank $0$
Graph

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

Elliptic curves in class 90.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
90.b1 90b4 \([1, -1, 1, -218, -269]\) \(57960603/31250\) \(615093750\) \([2]\) \(48\) \(0.37752\)  
90.b2 90b2 \([1, -1, 1, -128, 587]\) \(8527173507/200\) \(5400\) \([6]\) \(16\) \(-0.17179\)  
90.b3 90b1 \([1, -1, 1, -8, 11]\) \(-1860867/320\) \(-8640\) \([6]\) \(8\) \(-0.51836\) \(\Gamma_0(N)\)-optimal
90.b4 90b3 \([1, -1, 1, 52, -53]\) \(804357/500\) \(-9841500\) \([2]\) \(24\) \(0.030944\)  

Rank

sage: E.rank()
 

The elliptic curves in class 90.b have rank \(0\).

Complex multiplication

The elliptic curves in class 90.b do not have complex multiplication.

Modular form 90.2.a.b

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

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.