↑

Properties

Label 77910n
Number of curves $4$
Conductor $77910$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 0, 171818, 85724164]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, 171818, 85724164]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, 171818, 85724164]) 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 77910n have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(5\)\(1 - T\)
\(7\)\(1\)
\(53\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 5 T + 11 T^{2}\) 1.11.f
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 4 T + 17 T^{2}\) 1.17.ae
\(19\) \( 1 + T + 19 T^{2}\) 1.19.b
\(23\) \( 1 + 5 T + 23 T^{2}\) 1.23.f
\(29\) \( 1 + T + 29 T^{2}\) 1.29.b
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 77910n do not have complex multiplication.

Modular form 77910.2.a.n

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);
 
\(q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} - q^{10} - q^{12} + 2 q^{13} - q^{15} + q^{16} - 2 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

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

Elliptic curves in class 77910n

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
77910.r4 77910n1 \([1, 1, 0, 171818, 85724164]\) \(4768013769464231/29697948831600\) \(-3493933982088908400\) \([2]\) \(1474560\) \(2.2400\) \(\Gamma_0(N)\)-optimal
77910.r3 77910n2 \([1, 1, 0, -2181162, 1124329536]\) \(9754377335041367449/995626517602500\) \(117134464169416522500\) \([2, 2]\) \(2949120\) \(2.5865\)  
77910.r2 77910n3 \([1, 1, 0, -8015592, -7507126206]\) \(484108118865316036729/73399966614843750\) \(8635432672269752343750\) \([2]\) \(5898240\) \(2.9331\)  
77910.r1 77910n4 \([1, 1, 0, -33994412, 76273588686]\) \(36928196050908253259449/452758954469850\) \(53266638234423382650\) \([2]\) \(5898240\) \(2.9331\)