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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 1, -2096438, -1167357819]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, -2096438, -1167357819]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, -2096438, -1167357819]) 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 31410.e have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1 + T\)
\(349\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 31410.e do not have complex multiplication.

Modular form 31410.2.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);
 
\(q + q^{2} + q^{4} - q^{5} - 4 q^{7} + q^{8} - q^{10} + 4 q^{11} - 2 q^{13} - 4 q^{14} + q^{16} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

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

Elliptic curves in class 31410.e

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
31410.e1 31410g4 \([1, -1, 1, -2096438, -1167357819]\) \(1397790417785316543001/640892891563200\) \(467210917949572800\) \([2]\) \(995328\) \(2.3474\)  
31410.e2 31410g3 \([1, -1, 1, -1157558, 471417477]\) \(235301185027835613721/4636822725000000\) \(3380243766525000000\) \([2]\) \(995328\) \(2.3474\)  
31410.e3 31410g2 \([1, -1, 1, -152438, -11844219]\) \(537369779909439001/227309898240000\) \(165708915816960000\) \([2, 2]\) \(497664\) \(2.0009\)  
31410.e4 31410g1 \([1, -1, 1, 31882, -1374843]\) \(4916382075769319/3952292659200\) \(-2881221348556800\) \([4]\) \(248832\) \(1.6543\) \(\Gamma_0(N)\)-optimal