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Properties

Label 52983a
Number of curves $6$
Conductor $52983$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("a1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 52983a have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1\)
\(7\)\(1 - T\)
\(29\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + 2 T + 2 T^{2}\) 1.2.c
\(5\) \( 1 - 4 T + 5 T^{2}\) 1.5.ae
\(11\) \( 1 - 2 T + 11 T^{2}\) 1.11.ac
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 5 T + 19 T^{2}\) 1.19.f
\(23\) \( 1 + 9 T + 23 T^{2}\) 1.23.j
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 52983a do not have complex multiplication.

Modular form 52983.2.a.a

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{4} + 2 q^{5} - q^{7} + 3 q^{8} - 2 q^{10} + 4 q^{11} - 2 q^{13} + q^{14} - q^{16} - 6 q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 52983a

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
52983.e6 52983a1 \([1, -1, 1, 7411, 52260]\) \(103823/63\) \(-27318450663567\) \([2]\) \(100352\) \(1.2674\) \(\Gamma_0(N)\)-optimal
52983.e5 52983a2 \([1, -1, 1, -30434, 445848]\) \(7189057/3969\) \(1721062391804721\) \([2, 2]\) \(200704\) \(1.6140\)  
52983.e3 52983a3 \([1, -1, 1, -295349, -61332330]\) \(6570725617/45927\) \(19915150533740343\) \([2]\) \(401408\) \(1.9606\)  
52983.e2 52983a4 \([1, -1, 1, -371039, 86959518]\) \(13027640977/21609\) \(9370228577603481\) \([2, 2]\) \(401408\) \(1.9606\)  
52983.e4 52983a5 \([1, -1, 1, -257504, 141093006]\) \(-4354703137/17294403\) \(-7499306271608652627\) \([2]\) \(802816\) \(2.3072\)  
52983.e1 52983a6 \([1, -1, 1, -5934254, 5565613650]\) \(53297461115137/147\) \(63743051548323\) \([2]\) \(802816\) \(2.3072\)