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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 1, -6031721, 5701270380]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -6031721, 5701270380]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -6031721, 5701270380]) 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 30926.a have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(7\)\(1 - T\)
\(47\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 2 T + 3 T^{2}\) 1.3.c
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 2 T + 19 T^{2}\) 1.19.c
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 30926.a do not have complex multiplication.

Modular form 30926.2.a.a

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);
 
\(q - q^{2} - 2 q^{3} + q^{4} + 2 q^{6} + q^{7} - q^{8} + q^{9} - 2 q^{12} + 4 q^{13} - q^{14} + q^{16} + 6 q^{17} - q^{18} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 30926.a

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
30926.a1 30926f6 \([1, 0, 1, -6031721, 5701270380]\) \(2251439055699625/25088\) \(270428954173952\) \([2]\) \(596160\) \(2.3382\)  
30926.a2 30926f5 \([1, 0, 1, -376681, 89208684]\) \(-548347731625/1835008\) \(-19779946362437632\) \([2]\) \(298080\) \(1.9916\)  
30926.a3 30926f4 \([1, 0, 1, -78466, 6927852]\) \(4956477625/941192\) \(10145311233932168\) \([2]\) \(198720\) \(1.7889\)  
30926.a4 30926f2 \([1, 0, 1, -23241, -1364734]\) \(128787625/98\) \(1056363102242\) \([2]\) \(66240\) \(1.2396\)  
30926.a5 30926f1 \([1, 0, 1, -1151, -30498]\) \(-15625/28\) \(-301818029212\) \([2]\) \(33120\) \(0.89299\) \(\Gamma_0(N)\)-optimal
30926.a6 30926f3 \([1, 0, 1, 9894, 636620]\) \(9938375/21952\) \(-236625334902208\) \([2]\) \(99360\) \(1.4423\)