↑

Properties

Label 35574bg
Number of curves $6$
Conductor $35574$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 1, -23840, 3126638]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -23840, 3126638]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -23840, 3126638]) 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 35574bg have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 - T\)
\(7\)\(1\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - T + 5 T^{2}\) 1.5.ab
\(13\) \( 1 - 3 T + 13 T^{2}\) 1.13.ad
\(17\) \( 1 + T + 17 T^{2}\) 1.17.b
\(19\) \( 1 + 8 T + 19 T^{2}\) 1.19.i
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 9 T + 29 T^{2}\) 1.29.aj
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 35574bg do not have complex multiplication.

Modular form 35574.2.a.bg

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

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

Isogeny graph

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

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

Elliptic curves in class 35574bg

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
35574.bj5 35574bg1 \([1, 0, 1, -23840, 3126638]\) \(-7189057/16128\) \(-3361436146075392\) \([2]\) \(245760\) \(1.6679\) \(\Gamma_0(N)\)-optimal
35574.bj4 35574bg2 \([1, 0, 1, -498160, 135177326]\) \(65597103937/63504\) \(13235654825171856\) \([2, 2]\) \(491520\) \(2.0144\)  
35574.bj3 35574bg3 \([1, 0, 1, -616740, 65926606]\) \(124475734657/63011844\) \(13133078500276774116\) \([2, 2]\) \(983040\) \(2.3610\)  
35574.bj1 35574bg4 \([1, 0, 1, -7968700, 8657569358]\) \(268498407453697/252\) \(52522439782428\) \([2]\) \(983040\) \(2.3610\)  
35574.bj6 35574bg5 \([1, 0, 1, 2288470, 509842694]\) \(6359387729183/4218578658\) \(-879246204493019540562\) \([2]\) \(1966080\) \(2.7076\)  
35574.bj2 35574bg6 \([1, 0, 1, -5419230, -4809561242]\) \(84448510979617/933897762\) \(194645194315830460818\) \([2]\) \(1966080\) \(2.7076\)