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Properties

Label 80223.j
Number of curves $6$
Conductor $80223$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("j1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 80223.j have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1 + T\)
\(11\)\(1\)
\(13\)\(1 + T\)
\(17\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 80223.j do not have complex multiplication.

Modular form 80223.2.a.j

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 80223.j

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
80223.j1 80223c6 \([1, 1, 0, -2441056, 1466719159]\) \(908031902324522977/161726530797\) \(286508414625264117\) \([2]\) \(1310720\) \(2.3541\)  
80223.j2 80223c4 \([1, 1, 0, -168071, 17918520]\) \(296380748763217/92608836489\) \(164062202979289329\) \([2, 2]\) \(655360\) \(2.0075\)  
80223.j3 80223c2 \([1, 1, 0, -65826, -6313545]\) \(17806161424897/668584449\) \(1184438135054889\) \([2, 2]\) \(327680\) \(1.6610\)  
80223.j4 80223c1 \([1, 1, 0, -65221, -6438296]\) \(17319700013617/25857\) \(45807252777\) \([2]\) \(163840\) \(1.3144\) \(\Gamma_0(N)\)-optimal
80223.j5 80223c3 \([1, 1, 0, 26739, -22549446]\) \(1193377118543/124806800313\) \(-221102859969298593\) \([2]\) \(655360\) \(2.0075\)  
80223.j6 80223c5 \([1, 1, 0, 468994, 122014941]\) \(6439735268725823/7345472585373\) \(-13012952758815977253\) \([2]\) \(1310720\) \(2.3541\)