Properties

Label 77.c
Number of curves $2$
Conductor $77$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("c1")
 
E.isogeny_class()
 

Elliptic curves in class 77.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
77.c1 77c2 \([1, 1, 0, -51, 110]\) \(15124197817/1294139\) \(1294139\) \([2]\) \(12\) \(-0.086020\)  
77.c2 77c1 \([1, 1, 0, 4, 11]\) \(4657463/41503\) \(-41503\) \([2]\) \(6\) \(-0.43259\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 77.c have rank \(0\).

Complex multiplication

The elliptic curves in class 77.c do not have complex multiplication.

Modular form 77.2.a.c

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