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Properties

Label 23184bn
Number of curves $4$
Conductor $23184$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 0, 668805, 1482636458]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, 668805, 1482636458]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, 668805, 1482636458]) 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 23184bn have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(7\)\(1 + T\)
\(23\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(11\) \( 1 - 3 T + 11 T^{2}\) 1.11.ad
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 - 8 T + 17 T^{2}\) 1.17.ai
\(19\) \( 1 - 5 T + 19 T^{2}\) 1.19.af
\(29\) \( 1 + 8 T + 29 T^{2}\) 1.29.i
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 23184bn do not have complex multiplication.

Modular form 23184.2.a.bn

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);
 
\(q - q^{7} + 6 q^{11} + 2 q^{13} + 6 q^{17} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

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 23184bn

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
23184.bb4 23184bn1 \([0, 0, 0, 668805, 1482636458]\) \(11079872671250375/324440155855872\) \(-968773114343140098048\) \([2]\) \(921600\) \(2.7069\) \(\Gamma_0(N)\)-optimal
23184.bb2 23184bn2 \([0, 0, 0, -16127355, 23750985386]\) \(155355156733986861625/8291568305839392\) \(24758490296143531081728\) \([2]\) \(1843200\) \(3.0535\)  
23184.bb3 23184bn3 \([0, 0, 0, -6037995, -40734523366]\) \(-8152944444844179625/235342826399858688\) \(-702729914144755644628992\) \([2]\) \(2764800\) \(3.2562\)  
23184.bb1 23184bn4 \([0, 0, 0, -218374635, -1236062404582]\) \(385693937170561837203625/2159357734550274048\) \(6447807645643365502943232\) \([2]\) \(5529600\) \(3.6028\)