Properties

Label 3757b
Number of curves $2$
Conductor $3757$
CM no
Rank $1$
Graph

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

Elliptic curves in class 3757b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3757.b2 3757b1 \([1, -1, 1, -211747, 37495370]\) \(43499078731809/82055753\) \(1980626399884457\) \([2]\) \(34560\) \(1.8252\) \(\Gamma_0(N)\)-optimal
3757.b1 3757b2 \([1, -1, 1, -3386412, 2399446130]\) \(177930109857804849/634933\) \(15325739097877\) \([2]\) \(69120\) \(2.1718\)  

Rank

sage: E.rank()
 

The elliptic curves in class 3757b have rank \(1\).

Complex multiplication

The elliptic curves in class 3757b do not have complex multiplication.

Modular form 3757.2.a.b

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