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Properties

Label 121296.cp
Number of curves $6$
Conductor $121296$
CM no
Rank $2$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cp1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 121296.cp have rank \(2\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(7\)\(1 + T\)
\(19\)\(1\)
 
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.g
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 121296.cp do not have complex multiplication.

Modular form 121296.2.a.cp

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} - q^{7} + q^{9} - 6 q^{11} + 4 q^{13} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.

Elliptic curves in class 121296.cp

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
121296.cp1 121296cn6 \([0, 1, 0, -56528039968, -5173037018309644]\) \(103665426767620308239307625/5961940992\) \(1148865603332726587392\) \([2]\) \(179159040\) \(4.4276\)  
121296.cp2 121296cn5 \([0, 1, 0, -3533008928, -80829277725708]\) \(25309080274342544331625/191933498523648\) \(36985571456848771046965248\) \([2]\) \(89579520\) \(4.0810\)  
121296.cp3 121296cn4 \([0, 1, 0, -698523568, -7082473116076]\) \(195607431345044517625/752875610010048\) \(145079076275548680242331648\) \([2]\) \(59719680\) \(3.8783\)  
121296.cp4 121296cn3 \([0, 1, 0, -64549808, 6114289236]\) \(154357248921765625/89242711068672\) \(17197064048861699031171072\) \([2]\) \(29859840\) \(3.5317\)  
121296.cp5 121296cn2 \([0, 1, 0, -45864448, 111994688180]\) \(55369510069623625/3916046302812\) \(754621842852181267955712\) \([2]\) \(19906560\) \(3.3290\)  
121296.cp6 121296cn1 \([0, 1, 0, -45055808, 116390131764]\) \(52492168638015625/293197968\) \(56499227492228333568\) \([2]\) \(9953280\) \(2.9824\) \(\Gamma_0(N)\)-optimal