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Properties

Label 363888ck
Number of curves $6$
Conductor $363888$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ck1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 363888ck have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(7\)\(1 + T\)
\(19\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 - 6 T + 11 T^{2}\) 1.11.ag
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 363888ck do not have complex multiplication.

Modular form 363888.2.a.ck

Copy content sage:E.q_eigenform(10)
 
\(q - q^{7} + 6 q^{11} + 4 q^{13} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 363888ck

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
363888.ck6 363888ck1 \([0, 0, 0, -405502275, -3142939059902]\) \(52492168638015625/293197968\) \(41187936841834455171072\) \([2]\) \(79626240\) \(3.5317\) \(\Gamma_0(N)\)-optimal
363888.ck5 363888ck2 \([0, 0, 0, -412780035, -3024269360894]\) \(55369510069623625/3916046302812\) \(550119323439240144339714048\) \([2]\) \(159252480\) \(3.8783\)  
363888.ck4 363888ck3 \([0, 0, 0, -580948275, -165666757646]\) \(154357248921765625/89242711068672\) \(12536659691620178593723711488\) \([2]\) \(238878720\) \(4.0810\)  
363888.ck3 363888ck4 \([0, 0, 0, -6286712115, 191220487421938]\) \(195607431345044517625/752875610010048\) \(105762646604874987896659771392\) \([2]\) \(477757440\) \(4.4276\)  
363888.ck2 363888ck5 \([0, 0, 0, -31797080355, 2182358701513762]\) \(25309080274342544331625/191933498523648\) \(26962481592042754093237665792\) \([2]\) \(716636160\) \(4.6303\)  
363888.ck1 363888ck6 \([0, 0, 0, -508752359715, 139671490742000674]\) \(103665426767620308239307625/5961940992\) \(837523024829557682208768\) \([2]\) \(1433272320\) \(4.9769\)