Properties

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

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 3757a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3757.d1 3757a1 \([1, 0, 1, -17202, 866831]\) \(23320116793/2873\) \(69347235737\) \([2]\) \(6912\) \(1.1036\) \(\Gamma_0(N)\)-optimal
3757.d2 3757a2 \([1, 0, 1, -15757, 1018845]\) \(-17923019113/8254129\) \(-199234608272401\) \([2]\) \(13824\) \(1.4502\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 3757.2.a.a

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