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Properties

Label 60984cg
Number of curves $6$
Conductor $60984$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cg1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 60984cg have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(7\)\(1 - T\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 3 T + 5 T^{2}\) 1.5.ad
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 - 4 T + 17 T^{2}\) 1.17.ae
\(19\) \( 1 + 8 T + 19 T^{2}\) 1.19.i
\(23\) \( 1 - 3 T + 23 T^{2}\) 1.23.ad
\(29\) \( 1 - 4 T + 29 T^{2}\) 1.29.ae
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 60984cg do not have complex multiplication.

Modular form 60984.2.a.cg

Copy content sage:E.q_eigenform(10)
 
\(q + 2 q^{5} + q^{7} + 2 q^{13} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 60984cg

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
60984.bz4 60984cg1 \([0, 0, 0, -2264394, 1311523477]\) \(62140690757632/6237\) \(128878171562448\) \([2]\) \(737280\) \(2.1388\) \(\Gamma_0(N)\)-optimal
60984.bz3 60984cg2 \([0, 0, 0, -2269839, 1304899090]\) \(3911877700432/38900169\) \(12861010496559810816\) \([2, 2]\) \(1474560\) \(2.4854\)  
60984.bz5 60984cg3 \([0, 0, 0, -592779, 3190249942]\) \(-17418812548/3314597517\) \(-4383433239892411364352\) \([2]\) \(2949120\) \(2.8320\)  
60984.bz2 60984cg4 \([0, 0, 0, -4034019, -1004412530]\) \(5489767279588/2847396321\) \(3765576851066820658176\) \([2, 2]\) \(2949120\) \(2.8320\)  
60984.bz6 60984cg5 \([0, 0, 0, 15175941, -7808580362]\) \(146142660369886/94532266521\) \(-250030887421629574858752\) \([2]\) \(5898240\) \(3.1786\)  
60984.bz1 60984cg6 \([0, 0, 0, -51470859, -141996188378]\) \(5701568801608514/6277868289\) \(16604499576513198163968\) \([2]\) \(5898240\) \(3.1786\)