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Properties

Label 77658p
Number of curves $2$
Conductor $77658$
CM no
Rank $1$
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Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 77658p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
77658.q2 77658p1 \([1, 0, 1, 12904, 36552854]\) \(37595375/91325808\) \(-577303588111268592\) \([2]\) \(1064448\) \(2.0870\) \(\Gamma_0(N)\)-optimal
77658.q1 77658p2 \([1, 0, 1, -1577236, 746391350]\) \(68644006908625/1639085868\) \(10361256840113291532\) \([2]\) \(2128896\) \(2.4336\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 77658.2.a.p

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