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Properties

Label 221760.jo
Number of curves $6$
Conductor $221760$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("jo1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 221760.jo have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(7\)\(1 + T\)
\(11\)\(1 - T\)
 
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.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 221760.jo do not have complex multiplication.

Modular form 221760.2.a.jo

Copy content sage:E.q_eigenform(10)
 
\(q + q^{5} - q^{7} + q^{11} + 2 q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 221760.jo

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
221760.jo1 221760jk5 \([0, 0, 0, -1095324492, 13952830285424]\) \(1520949008089505953959842/278553515625\) \(26616202905600000000\) \([2]\) \(37748736\) \(3.5632\)  
221760.jo2 221760jk3 \([0, 0, 0, -68464812, 217965949616]\) \(742879737792994384804/317817082130625\) \(15183947554699714560000\) \([2, 2]\) \(18874368\) \(3.2166\)  
221760.jo3 221760jk6 \([0, 0, 0, -57772812, 288349247216]\) \(-223180773010681046402/246754509479287425\) \(-23577760551456018638438400\) \([2]\) \(37748736\) \(3.5632\)  
221760.jo4 221760jk2 \([0, 0, 0, -4954332, 2258955344]\) \(1125982298608534096/467044181552025\) \(5578345813629767270400\) \([2, 2]\) \(9437184\) \(2.8700\)  
221760.jo5 221760jk1 \([0, 0, 0, -2318952, -1334648824]\) \(1847444944806639616/38285567941005\) \(28580023325688468480\) \([2]\) \(4718592\) \(2.5234\) \(\Gamma_0(N)\)-optimal
221760.jo6 221760jk4 \([0, 0, 0, 16390068, 16542627824]\) \(10191978981888338876/8372623608979245\) \(-400008322150948510433280\) \([2]\) \(18874368\) \(3.2166\)