Properties

Label 51870.y
Number of curves $8$
Conductor $51870$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("y1")
 
E.isogeny_class()
 

Elliptic curves in class 51870.y

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51870.y1 51870bd8 \([1, 0, 1, -768208004, -8195392298998]\) \(50137213659805457275731367898809/4113897879000\) \(4113897879000\) \([2]\) \(7962624\) \(3.2723\)  
51870.y2 51870bd6 \([1, 0, 1, -48013004, -128055986998]\) \(12240533203187013248735018809/3506282465049000000\) \(3506282465049000000\) \([2, 2]\) \(3981312\) \(2.9257\)  
51870.y3 51870bd7 \([1, 0, 1, -47818004, -129147674998]\) \(-12091997009671629064982138809/207252595706436249879000\) \(-207252595706436249879000\) \([2]\) \(7962624\) \(3.2723\)  
51870.y4 51870bd5 \([1, 0, 1, -9485039, -11240205244]\) \(94371532824107026279203049/40995077600666342790\) \(40995077600666342790\) \([6]\) \(2654208\) \(2.7230\)  
51870.y5 51870bd3 \([1, 0, 1, -3013004, -1983986998]\) \(3024980849878413455018809/50557689000000000000\) \(50557689000000000000\) \([2]\) \(1990656\) \(2.5791\)  
51870.y6 51870bd2 \([1, 0, 1, -687089, -116077264]\) \(35872512095393194378249/14944558319037792900\) \(14944558319037792900\) \([2, 6]\) \(1327104\) \(2.3764\)  
51870.y7 51870bd1 \([1, 0, 1, -322589, 69234536]\) \(3712533999213317890249/76090919904090000\) \(76090919904090000\) \([6]\) \(663552\) \(2.0298\) \(\Gamma_0(N)\)-optimal
51870.y8 51870bd4 \([1, 0, 1, 2278861, -850446484]\) \(1308812680909424992398551/1070002284841633041990\) \(-1070002284841633041990\) \([6]\) \(2654208\) \(2.7230\)  

Rank

sage: E.rank()
 

The elliptic curves in class 51870.y have rank \(1\).

Complex multiplication

The elliptic curves in class 51870.y do not have complex multiplication.

Modular form 51870.2.a.y

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

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.