Properties

Label 57038n
Number of curves $3$
Conductor $57038$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("n1")
 
E.isogeny_class()
 

Elliptic curves in class 57038n

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
57038.k3 57038n1 \([1, 1, 1, -16794, -844661]\) \(11134383337/316\) \(14866498396\) \([]\) \(95040\) \(1.0535\) \(\Gamma_0(N)\)-optimal
57038.k2 57038n2 \([1, 1, 1, -29429, 570459]\) \(59914169497/31554496\) \(1484509063830976\) \([]\) \(285120\) \(1.6028\)  
57038.k1 57038n3 \([1, 1, 1, -1883164, 993887229]\) \(15698803397448457/20709376\) \(974290838880256\) \([]\) \(855360\) \(2.1521\)  

Rank

sage: E.rank()
 

The elliptic curves in class 57038n have rank \(1\).

Complex multiplication

The elliptic curves in class 57038n do not have complex multiplication.

Modular form 57038.2.a.n

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.