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Properties

Label 216384dt
Number of curves $4$
Conductor $216384$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("dt1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 216384dt have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(7\)\(1\)
\(23\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(11\) \( 1 - 5 T + 11 T^{2}\) 1.11.af
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 5 T + 19 T^{2}\) 1.19.af
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 216384dt do not have complex multiplication.

Modular form 216384.2.a.dt

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + q^{9} + 6 q^{11} + 2 q^{13} + 6 q^{17} - 2 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 216384dt

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
216384.ct4 216384dt1 \([0, -1, 0, 14565087, 150674939073]\) \(11079872671250375/324440155855872\) \(-10006052181452386448965632\) \([2]\) \(44236800\) \(3.4772\) \(\Gamma_0(N)\)-optimal
216384.ct2 216384dt2 \([0, -1, 0, -351217953, 2413920920769]\) \(155355156733986861625/8291568305839392\) \(255720087778413413507530752\) \([2]\) \(88473600\) \(3.8237\)  
216384.ct3 216384dt3 \([0, -1, 0, -131494113, -4139790691455]\) \(-8152944444844179625/235342826399858688\) \(-7258203274115016237911113728\) \([2]\) \(132710400\) \(4.0265\)  
216384.ct1 216384dt4 \([0, -1, 0, -4755714273, -125618979138687]\) \(385693937170561837203625/2159357734550274048\) \(66596707529419703313537957888\) \([2]\) \(265420800\) \(4.3730\)