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Properties

Label 5082.d
Number of curves $6$
Conductor $5082$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("d1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 5082.d have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(7\)\(1 - T\)
\(11\)\(1\)
 
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.g
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 5082.d do not have complex multiplication.

Modular form 5082.2.a.d

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - 2 q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + 2 q^{10} - q^{12} - 6 q^{13} - q^{14} + 2 q^{15} + q^{16} - 2 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content sage:E.isogeny_class().matrix()
 

The \(i,j\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 5082.d

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5082.d1 5082i3 \([1, 1, 0, -162626, -25310424]\) \(268498407453697/252\) \(446433372\) \([2]\) \(20480\) \(1.3880\)  
5082.d2 5082i5 \([1, 1, 0, -110596, 13974646]\) \(84448510979617/933897762\) \(1654456853146482\) \([2]\) \(40960\) \(1.7346\)  
5082.d3 5082i4 \([1, 1, 0, -12586, -197600]\) \(124475734657/63011844\) \(111629325368484\) \([2, 2]\) \(20480\) \(1.3880\)  
5082.d4 5082i2 \([1, 1, 0, -10166, -398460]\) \(65597103937/63504\) \(112501209744\) \([2, 2]\) \(10240\) \(1.0415\)  
5082.d5 5082i1 \([1, 1, 0, -486, -9324]\) \(-7189057/16128\) \(-28571735808\) \([2]\) \(5120\) \(0.69490\) \(\Gamma_0(N)\)-optimal
5082.d6 5082i6 \([1, 1, 0, 46704, -1466406]\) \(6359387729183/4218578658\) \(-7473469425945138\) \([2]\) \(40960\) \(1.7346\)