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Properties

Label 77658i
Number of curves $2$
Conductor $77658$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 77658i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
77658.m2 77658i1 \([1, 1, 0, 18573167, -14758612715]\) \(1409825840597/1003290624\) \(-504246455247905062060032\) \([2]\) \(15257088\) \(3.2365\) \(\Gamma_0(N)\)-optimal
77658.m1 77658i2 \([1, 1, 0, -83195793, -124567320555]\) \(126710241047083/59997563136\) \(30154331966367885137019648\) \([2]\) \(30514176\) \(3.5831\)  

Rank

sage: E.rank()
 

The elliptic curves in class 77658i have rank \(0\).

Complex multiplication

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

Modular form 77658.2.a.i

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + 4 q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - 4 q^{10} + 2 q^{11} - q^{12} + 2 q^{13} - q^{14} - 4 q^{15} + q^{16} + 2 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.