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Properties

Label 77658y
Number of curves $3$
Conductor $77658$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("y1")
 
E.isogeny_class()
 

Elliptic curves in class 77658y

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
77658.v3 77658y1 \([1, 1, 1, -289407, -58581513]\) \(424072554697/11849166\) \(74902880113867134\) \([]\) \(1330560\) \(2.0171\) \(\Gamma_0(N)\)-optimal
77658.v2 77658y2 \([1, 1, 1, -3035172, 2012723757]\) \(489173485343257/5890514616\) \(37236081433176824184\) \([]\) \(3991680\) \(2.5664\)  
77658.v1 77658y3 \([1, 1, 1, -245133987, 1477145848257]\) \(257705427598877502217/462336\) \(2922593706622464\) \([]\) \(11975040\) \(3.1157\)  

Rank

sage: E.rank()
 

The elliptic curves in class 77658y have rank \(1\).

Complex multiplication

The elliptic curves in class 77658y do not have complex multiplication.

Modular form 77658.2.a.y

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - 3 q^{5} - q^{6} - q^{7} + q^{8} + q^{9} - 3 q^{10} - q^{12} + 5 q^{13} - q^{14} + 3 q^{15} + q^{16} + 3 q^{17} + q^{18} + 7 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.