Properties

Label 1045.b
Number of curves $4$
Conductor $1045$
CM no
Rank $1$
Graph

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

Elliptic curves in class 1045.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1045.b1 1045b3 \([1, -1, 0, -1016789, -346894930]\) \(116256292809537371612841/15216540068579856875\) \(15216540068579856875\) \([4]\) \(17664\) \(2.4090\)  
1045.b2 1045b2 \([1, -1, 0, -982414, -374539305]\) \(104859453317683374662841/2223652969140625\) \(2223652969140625\) \([2, 2]\) \(8832\) \(2.0624\)  
1045.b3 1045b1 \([1, -1, 0, -982409, -374543312]\) \(104857852278310619039721/47155625\) \(47155625\) \([2]\) \(4416\) \(1.7158\) \(\Gamma_0(N)\)-optimal
1045.b4 1045b4 \([1, -1, 0, -948119, -401927292]\) \(-94256762600623910012361/15323275604248046875\) \(-15323275604248046875\) \([2]\) \(17664\) \(2.4090\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1045.b have rank \(1\).

Complex multiplication

The elliptic curves in class 1045.b do not have complex multiplication.

Modular form 1045.2.a.b

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.