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Properties

Label 350658.dw
Number of curves $6$
Conductor $350658$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 1, -86176949, -307893151695]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, -86176949, -307893151695]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, -86176949, -307893151695]) 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 350658.dw have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 350658.dw do not have complex multiplication.

Modular form 350658.2.a.dw

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

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

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 350658.dw

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
350658.dw1 350658dw6 \([1, -1, 1, -86176949, -307893151695]\) \(54804145548726848737/637608031452\) \(823450349397402560988\) \([2]\) \(41943040\) \(3.1642\)  
350658.dw2 350658dw3 \([1, -1, 1, -19290569, 32615063385]\) \(614716917569296417/19093020912\) \(24658024939295167728\) \([2]\) \(20971520\) \(2.8177\)  
350658.dw3 350658dw4 \([1, -1, 1, -5525609, -4547331687]\) \(14447092394873377/1439452851984\) \(1859007251223034100496\) \([2, 2]\) \(20971520\) \(2.8177\)  
350658.dw4 350658dw2 \([1, -1, 1, -1256729, 464333433]\) \(169967019783457/26337394944\) \(34013901957078548736\) \([2, 2]\) \(10485760\) \(2.4711\)  
350658.dw5 350658dw1 \([1, -1, 1, 137191, 40024185]\) \(221115865823/664731648\) \(-858479631372582912\) \([2]\) \(5242880\) \(2.1245\) \(\Gamma_0(N)\)-optimal
350658.dw6 350658dw5 \([1, -1, 1, 6823651, -21999305919]\) \(27207619911317663/177609314617308\) \(-229376740824296775007452\) \([2]\) \(41943040\) \(3.1642\)