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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 0, -2747943, 1769945013]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -2747943, 1769945013]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -2747943, 1769945013]) 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 curve 5070.b1 has rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(5\)\(1 + T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 3 T + 7 T^{2}\) 1.7.ad
\(11\) \( 1 + 3 T + 11 T^{2}\) 1.11.d
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 - 3 T + 19 T^{2}\) 1.19.ad
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 + 4 T + 29 T^{2}\) 1.29.e
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 5070.b do not have complex multiplication.

Modular form 5070.2.a.b

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);
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + 3 q^{7} - q^{8} + q^{9} + q^{10} - 3 q^{11} - q^{12} - 3 q^{14} + q^{15} + q^{16} - q^{18} + 3 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny graph

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

Elliptic curves in class 5070.b

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
5070.b1 5070c1 \([1, 1, 0, -2747943, 1769945013]\) \(-2813198004118489/33177600000\) \(-27063987569049600000\) \([]\) \(212160\) \(2.5413\) \(\Gamma_0(N)\)-optimal