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Properties

Label 8670e
Number of curves $8$
Conductor $8670$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 0, 428, 10624]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, 428, 10624]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, 428, 10624]) 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 8670e have rank \(0\).

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 \(\mathbb{F}_p\)
\(7\) \( 1 + 3 T + 7 T^{2}\) 1.7.d
\(11\) \( 1 + 3 T + 11 T^{2}\) 1.11.d
\(13\) \( 1 + T + 13 T^{2}\) 1.13.b
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + T + 23 T^{2}\) 1.23.b
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 8670.2.a.e

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);
 
\(q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + 4 q^{7} - q^{8} + q^{9} - q^{10} - q^{12} + 2 q^{13} - 4 q^{14} - q^{15} + q^{16} - q^{18} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

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 8670e

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.g8 8670e1 \([1, 1, 0, 428, 10624]\) \(357911/2160\) \(-52137149040\) \([2]\) \(9216\) \(0.73832\) \(\Gamma_0(N)\)-optimal
8670.g6 8670e2 \([1, 1, 0, -5352, 134316]\) \(702595369/72900\) \(1759628780100\) \([2, 2]\) \(18432\) \(1.0849\)  
8670.g7 8670e3 \([1, 1, 0, -3907, -309299]\) \(-273359449/1536000\) \(-37075305984000\) \([2]\) \(27648\) \(1.2876\)  
8670.g5 8670e4 \([1, 1, 0, -19802, -932094]\) \(35578826569/5314410\) \(128276938069290\) \([2]\) \(36864\) \(1.4315\)  
8670.g4 8670e5 \([1, 1, 0, -83382, 9232614]\) \(2656166199049/33750\) \(814642953750\) \([2]\) \(36864\) \(1.4315\)  
8670.g3 8670e6 \([1, 1, 0, -96387, -11536371]\) \(4102915888729/9000000\) \(217238121000000\) \([2, 2]\) \(55296\) \(1.6342\)  
8670.g1 8670e7 \([1, 1, 0, -1541387, -737215371]\) \(16778985534208729/81000\) \(1955143089000\) \([2]\) \(110592\) \(1.9808\)  
8670.g2 8670e8 \([1, 1, 0, -131067, -2540379]\) \(10316097499609/5859375000\) \(141431068359375000\) \([2]\) \(110592\) \(1.9808\)