↑

Properties

Label 394485q
Number of curves $6$
Conductor $394485$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("q1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 394485q have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(7\)\(1 + T\)
\(13\)\(1 - T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + T + 2 T^{2}\) 1.2.b
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 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 394485q do not have complex multiplication.

Modular form 394485.2.a.q

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - q^{7} + 3 q^{8} + q^{9} + q^{10} - 4 q^{11} - q^{12} + q^{13} + q^{14} - q^{15} - q^{16} - q^{18} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 394485q

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
394485.q5 394485q1 \([1, 0, 0, 43344, 19064511]\) \(373092501599/6718359375\) \(-162164862980859375\) \([2]\) \(3932160\) \(1.9827\) \(\Gamma_0(N)\)-optimal
394485.q4 394485q2 \([1, 0, 0, -859781, 289460136]\) \(2912015927948401/184878500625\) \(4462517565452480625\) \([2, 2]\) \(7864320\) \(2.3293\)  
394485.q2 394485q3 \([1, 0, 0, -13539656, 19174865961]\) \(11372424889583066401/50586128775\) \(1221026173749447975\) \([2]\) \(15728640\) \(2.6759\)  
394485.q3 394485q4 \([1, 0, 0, -2629906, -1287013189]\) \(83339496416030401/18593645841225\) \(448805409454131482025\) \([2, 2]\) \(15728640\) \(2.6759\)  
394485.q6 394485q5 \([1, 0, 0, 5917269, -7914492684]\) \(949279533867428399/1670570708285115\) \(-40323515740610834985435\) \([2]\) \(31457280\) \(3.0225\)  
394485.q1 394485q6 \([1, 0, 0, -39499081, -95546745994]\) \(282352188585428161201/20813369346315\) \(502384138719163208235\) \([2]\) \(31457280\) \(3.0225\)