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Properties

Label 870i
Number of curves $4$
Conductor $870$
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, 0, 0, -4480, -25600]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -4480, -25600]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -4480, -25600]) 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 870i have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(29\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(11\) \( 1 + 2 T + 11 T^{2}\) 1.11.c
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 870i do not have complex multiplication.

Modular form 870.2.a.i

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);
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - 2 q^{7} + q^{8} + q^{9} + q^{10} + 2 q^{11} + q^{12} + 4 q^{13} - 2 q^{14} + q^{15} + q^{16} - 2 q^{17} + q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

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 870i

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
870.h4 870i1 \([1, 0, 0, -4480, -25600]\) \(9944061759313921/5479747200000\) \(5479747200000\) \([10]\) \(1600\) \(1.1349\) \(\Gamma_0(N)\)-optimal
870.h3 870i2 \([1, 0, 0, -43360, 3450272]\) \(9015548596898711041/63863437500000\) \(63863437500000\) \([10]\) \(3200\) \(1.4815\)  
870.h2 870i3 \([1, 0, 0, -2136580, -1202240020]\) \(1078651622544688278688321/3692006820\) \(3692006820\) \([2]\) \(8000\) \(1.9396\)  
870.h1 870i4 \([1, 0, 0, -2136610, -1202204578]\) \(1078697059648930939019041/63106084995030150\) \(63106084995030150\) \([2]\) \(16000\) \(2.2862\)