Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 1, -7435, -545482]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -7435, -545482]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -7435, -545482]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

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

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 - T\)
\(7\)\(1 - T\)
\(43\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 77658.2.a.t

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q - q^{2} + q^{3} + q^{4} + 2 q^{5} - q^{6} + q^{7} - q^{8} + q^{9} - 2 q^{10} - 4 q^{11} + q^{12} + 6 q^{13} - q^{14} + 2 q^{15} + q^{16} + 2 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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.

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

Isogeny graph

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labeled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 77658t

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