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Properties

Label 194040.ds
Number of curves $6$
Conductor $194040$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ds1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 194040.ds have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 194040.ds do not have complex multiplication.

Modular form 194040.2.a.ds

Copy content sage:E.q_eigenform(10)
 
\(q + q^{5} - q^{11} + 2 q^{13} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.

Elliptic curves in class 194040.ds

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
194040.ds1 194040cx6 \([0, 0, 0, -13417725027, -598227598487554]\) \(1520949008089505953959842/278553515625\) \(48927650869389600000000\) \([2]\) \(113246208\) \(4.1895\)  
194040.ds2 194040cx4 \([0, 0, 0, -838693947, -9345290089786]\) \(742879737792994384804/317817082130625\) \(27912128841607292472960000\) \([2, 2]\) \(56623104\) \(3.8430\)  
194040.ds3 194040cx5 \([0, 0, 0, -707716947, -12362973974386]\) \(-223180773010681046402/246754509479287425\) \(-43342186736222642762400614400\) \([2]\) \(113246208\) \(4.1895\)  
194040.ds4 194040cx2 \([0, 0, 0, -60690567, -96852710374]\) \(1125982298608534096/467044181552025\) \(10254481353558257649926400\) \([2, 2]\) \(28311552\) \(3.4964\)  
194040.ds5 194040cx1 \([0, 0, 0, -28407162, 57223068329]\) \(1847444944806639616/38285567941005\) \(52537674441311291065680\) \([2]\) \(14155776\) \(3.1498\) \(\Gamma_0(N)\)-optimal
194040.ds6 194040cx3 \([0, 0, 0, 200778333, -709265167954]\) \(10191978981888338876/8372623608979245\) \(-735321548324014707874452480\) \([2]\) \(56623104\) \(3.8430\)