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Properties

Label 64350ej
Number of curves $4$
Conductor $64350$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ej1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 64350ej have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1\)
\(11\)\(1 + T\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 5 T + 19 T^{2}\) 1.19.f
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 + 5 T + 29 T^{2}\) 1.29.f
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 64350ej do not have complex multiplication.

Modular form 64350.2.a.ej

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + 4 q^{7} + q^{8} + q^{11} - q^{13} + 4 q^{14} + q^{16} + 2 q^{19} + O(q^{20})\) 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 64350ej

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
64350.ff4 64350ej1 \([1, -1, 1, 198546520, 19955525943147]\) \(75991146714893572533071/15147028085515223040000\) \(-172534116786571837440000000000\) \([2]\) \(77414400\) \(4.2893\) \(\Gamma_0(N)\)-optimal
64350.ff3 64350ej2 \([1, -1, 1, -9939053480, 370392082743147]\) \(9532597152396244075685450929/313550122650789880627200\) \(3571531865819153484019200000000\) \([2]\) \(154828800\) \(4.6358\)  
64350.ff2 64350ej3 \([1, -1, 1, -49909997480, 4292439534903147]\) \(-1207087636168285491836819264689/236446260657750000000000\) \(-2693270687804683593750000000000\) \([2]\) \(232243200\) \(4.8386\)  
64350.ff1 64350ej4 \([1, -1, 1, -798597497480, 274688417034903147]\) \(4944928228995290413834018379264689/189679641808585500000\) \(2160569669975919210937500000\) \([2]\) \(464486400\) \(5.1852\)