Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -35, 12840]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -35, 12840]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -35, 12840]) 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 25215h have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1 + T\)
\(5\)\(1 - T\)
\(41\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 - 4 T + 17 T^{2}\) 1.17.ae
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 8 T + 29 T^{2}\) 1.29.i
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 25215h do not have complex multiplication.

Modular form 25215.2.a.h

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);
 
\(q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} + 3 q^{8} + q^{9} - q^{10} + 4 q^{11} - q^{12} + 2 q^{13} + q^{15} - q^{16} - 2 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 25215h

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
25215.f7 25215h1 \([1, 0, 0, -35, 12840]\) \(-1/15\) \(-71251563615\) \([2]\) \(16640\) \(0.76136\) \(\Gamma_0(N)\)-optimal
25215.f6 25215h2 \([1, 0, 0, -8440, 293567]\) \(13997521/225\) \(1068773454225\) \([2, 2]\) \(33280\) \(1.1079\)  
25215.f5 25215h3 \([1, 0, 0, -16845, -390600]\) \(111284641/50625\) \(240474027200625\) \([2, 2]\) \(66560\) \(1.4545\)  
25215.f4 25215h4 \([1, 0, 0, -134515, 18977882]\) \(56667352321/15\) \(71251563615\) \([2]\) \(66560\) \(1.4545\)  
25215.f8 25215h5 \([1, 0, 0, 58800, -2917143]\) \(4733169839/3515625\) \(-16699585222265625\) \([2]\) \(133120\) \(1.8011\)  
25215.f2 25215h6 \([1, 0, 0, -226970, -41617125]\) \(272223782641/164025\) \(779135848130025\) \([2, 2]\) \(133120\) \(1.8011\)  
25215.f3 25215h7 \([1, 0, 0, -184945, -57494170]\) \(-147281603041/215233605\) \(-1022382059916218805\) \([2]\) \(266240\) \(2.1477\)  
25215.f1 25215h8 \([1, 0, 0, -3630995, -2663397180]\) \(1114544804970241/405\) \(1923792217605\) \([2]\) \(266240\) \(2.1477\)