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Properties

Label 114240dk
Number of curves $6$
Conductor $114240$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("dk1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 114240dk have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(5\)\(1 - T\)
\(7\)\(1 + T\)
\(17\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - T + 13 T^{2}\) 1.13.ab
\(19\) \( 1 + 2 T + 19 T^{2}\) 1.19.c
\(23\) \( 1 - 5 T + 23 T^{2}\) 1.23.af
\(29\) \( 1 + 4 T + 29 T^{2}\) 1.29.e
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 114240dk do not have complex multiplication.

Modular form 114240.2.a.dk

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} - q^{5} - q^{7} + q^{9} - 4 q^{11} - 6 q^{13} - q^{15} + q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 114240dk

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
114240.fq5 114240dk1 \([0, 1, 0, 195839, 32465759]\) \(3168685387909439/3563732336640\) \(-934211049656156160\) \([2]\) \(1769472\) \(2.1347\) \(\Gamma_0(N)\)-optimal
114240.fq4 114240dk2 \([0, 1, 0, -1114881, 304833375]\) \(584614687782041281/184812061593600\) \(48447373074392678400\) \([2, 2]\) \(3538944\) \(2.4812\)  
114240.fq3 114240dk3 \([0, 1, 0, -7033601, -6950333601]\) \(146796951366228945601/5397929064360000\) \(1415034716647587840000\) \([2, 2]\) \(7077888\) \(2.8278\)  
114240.fq2 114240dk4 \([0, 1, 0, -16167681, 25012499295]\) \(1782900110862842086081/328139630024640\) \(86019835173179228160\) \([2]\) \(7077888\) \(2.8278\)  
114240.fq6 114240dk5 \([0, 1, 0, 2758399, -24777648801]\) \(8854313460877886399/1016927675429790600\) \(-266581488547867027046400\) \([2]\) \(14155776\) \(3.1744\)  
114240.fq1 114240dk6 \([0, 1, 0, -111525121, -453359005345]\) \(585196747116290735872321/836876053125000\) \(219382036070400000000\) \([2]\) \(14155776\) \(3.1744\)