Properties

Label 51714.b
Number of curves $2$
Conductor $51714$
CM no
Rank $1$
Graph

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

Elliptic curves in class 51714.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51714.b1 51714k1 \([1, -1, 0, -3834, -57056]\) \(1771561/612\) \(2153471181732\) \([2]\) \(122880\) \(1.0683\) \(\Gamma_0(N)\)-optimal
51714.b2 51714k2 \([1, -1, 0, 11376, -406886]\) \(46268279/46818\) \(-164740545402498\) \([2]\) \(245760\) \(1.4148\)  

Rank

sage: E.rank()
 

The elliptic curves in class 51714.b have rank \(1\).

Complex multiplication

The elliptic curves in class 51714.b do not have complex multiplication.

Modular form 51714.2.a.b

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.