Properties

Label 57498e
Number of curves $2$
Conductor $57498$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 57498e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
57498.e2 57498e1 \([1, 1, 0, 3233550, -168288588]\) \(1457309849609375/848195776512\) \(-2176238303799100305408\) \([2]\) \(3939840\) \(2.7838\) \(\Gamma_0(N)\)-optimal
57498.e1 57498e2 \([1, 1, 0, -12975410, -1364509836]\) \(94162220003958625/54181012560192\) \(139013654792045316510528\) \([2]\) \(7879680\) \(3.1303\)  

Rank

sage: E.rank()
 

The elliptic curves in class 57498e have rank \(0\).

Complex multiplication

The elliptic curves in class 57498e do not have complex multiplication.

Modular form 57498.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} + q^{7} - q^{8} + q^{9} - 4 q^{11} - q^{12} + 4 q^{13} - q^{14} + q^{16} + 8 q^{17} - q^{18} + 2 q^{19} + O(q^{20})\) 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.