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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -17249, 870519]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -17249, 870519]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -17249, 870519]) 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 3234.t have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 3234.t do not have complex multiplication.

Modular form 3234.2.a.t

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

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

Isogeny graph

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

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

Elliptic curves in class 3234.t

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
3234.t1 3234t3 \([1, 0, 0, -17249, 870519]\) \(4824238966273/66\) \(7764834\) \([2]\) \(4608\) \(0.87800\)  
3234.t2 3234t2 \([1, 0, 0, -1079, 13509]\) \(1180932193/4356\) \(512479044\) \([2, 2]\) \(2304\) \(0.53143\)  
3234.t3 3234t4 \([1, 0, 0, -589, 25955]\) \(-192100033/2371842\) \(-279044839458\) \([2]\) \(4608\) \(0.87800\)  
3234.t4 3234t1 \([1, 0, 0, -99, -15]\) \(912673/528\) \(62118672\) \([2]\) \(1152\) \(0.18485\) \(\Gamma_0(N)\)-optimal