Properties

Label 53371.b
Number of curves $3$
Conductor $53371$
CM no
Rank $1$
Graph

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

Elliptic curves in class 53371.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
53371.b1 53371a3 \([0, -1, 1, -2161057, -1222057453]\) \(-50357871050752/19\) \(-421122861451\) \([]\) \(432432\) \(2.0186\)  
53371.b2 53371a2 \([0, -1, 1, -26217, -1729538]\) \(-89915392/6859\) \(-152025352983811\) \([]\) \(144144\) \(1.4693\)  
53371.b3 53371a1 \([0, -1, 1, 1873, -2003]\) \(32768/19\) \(-421122861451\) \([]\) \(48048\) \(0.91997\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 53371.b have rank \(1\).

Complex multiplication

The elliptic curves in class 53371.b do not have complex multiplication.

Modular form 53371.2.a.b

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.