Properties

Label 77658.t
Number of curves $6$
Conductor $77658$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("t1")
 
E.isogeny_class()
 

Elliptic curves in class 77658.t

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
77658.t1 77658t4 \([1, 0, 1, -2485095, -1508072162]\) \(268498407453697/252\) \(1592983488348\) \([2]\) \(1290240\) \(2.0697\)  
77658.t2 77658t6 \([1, 0, 1, -1690025, 837384338]\) \(84448510979617/933897762\) \(5903506804250596338\) \([2]\) \(2580480\) \(2.4163\)  
77658.t3 77658t3 \([1, 0, 1, -192335, -11506354]\) \(124475734657/63011844\) \(398320742310952356\) \([2, 2]\) \(1290240\) \(2.0697\)  
77658.t4 77658t2 \([1, 0, 1, -155355, -23561834]\) \(65597103937/63504\) \(401431839063696\) \([2, 2]\) \(645120\) \(1.7231\)  
77658.t5 77658t1 \([1, 0, 1, -7435, -545482]\) \(-7189057/16128\) \(-101950943254272\) \([2]\) \(322560\) \(1.3766\) \(\Gamma_0(N)\)-optimal
77658.t6 77658t5 \([1, 0, 1, 713675, -88698406]\) \(6359387729183/4218578658\) \(-26667167247981208242\) \([2]\) \(2580480\) \(2.4163\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 77658.2.a.t

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

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.