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Properties

Label 8670v
Number of curves $8$
Conductor $8670$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -23126, 1661220]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -23126, 1661220]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -23126, 1661220]) 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 8670v have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 8670v do not have complex multiplication.

Modular form 8670.2.a.v

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);
 
\(q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} - q^{10} - 4 q^{11} + q^{12} - 2 q^{13} - q^{15} + q^{16} + q^{18} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 8670v

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
8670.v6 8670v1 \([1, 0, 0, -23126, 1661220]\) \(-56667352321/16711680\) \(-403379329105920\) \([2]\) \(36864\) \(1.5171\) \(\Gamma_0(N)\)-optimal
8670.v5 8670v2 \([1, 0, 0, -393046, 94807076]\) \(278202094583041/16646400\) \(401803628601600\) \([2, 2]\) \(73728\) \(1.8636\)  
8670.v4 8670v3 \([1, 0, 0, -416166, 83020500]\) \(330240275458561/67652010000\) \(1632955059363690000\) \([2, 2]\) \(147456\) \(2.2102\)  
8670.v2 8670v4 \([1, 0, 0, -6288646, 6069408116]\) \(1139466686381936641/4080\) \(98481281520\) \([2]\) \(147456\) \(2.2102\)  
8670.v3 8670v5 \([1, 0, 0, -2086586, -1086607584]\) \(41623544884956481/2962701562500\) \(71512413391251562500\) \([2, 2]\) \(294912\) \(2.5568\)  
8670.v7 8670v6 \([1, 0, 0, 884334, 498400200]\) \(3168685387909439/6278181696900\) \(-151540043903460836100\) \([2]\) \(294912\) \(2.5568\)  
8670.v1 8670v7 \([1, 0, 0, -32792836, -72282118834]\) \(161572377633716256481/914742821250\) \(22079667965176541250\) \([2]\) \(589824\) \(2.9033\)  
8670.v8 8670v8 \([1, 0, 0, 1892944, -4740612030]\) \(31077313442863199/420227050781250\) \(-10143259433898925781250\) \([2]\) \(589824\) \(2.9033\)