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Properties

Label 45177.e
Number of curves $4$
Conductor $45177$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 45177.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
45177.e1 45177e4 \([1, 0, 0, -813899, -282671406]\) \(23239401850153/1625151\) \(4169692839312759\) \([2]\) \(612864\) \(2.0503\)  
45177.e2 45177e3 \([1, 0, 0, -279989, 53683680]\) \(946098541513/61847313\) \(158683284289789017\) \([2]\) \(612864\) \(2.0503\)  
45177.e3 45177e2 \([1, 0, 0, -54104, -3826641]\) \(6826561273/1490841\) \(3825090125319969\) \([2, 2]\) \(306432\) \(1.7037\)  
45177.e4 45177e1 \([1, 0, 0, 7501, -364440]\) \(18191447/32967\) \(-84584302525503\) \([4]\) \(153216\) \(1.3572\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 45177.e have rank \(0\).

Complex multiplication

The elliptic curves in class 45177.e do not have complex multiplication.

Modular form 45177.2.a.e

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.