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Properties

Label 189618x
Number of curves $4$
Conductor $189618$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("x1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 189618x have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 - T\)
\(11\)\(1 + T\)
\(13\)\(1\)
\(17\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 4 T + 5 T^{2}\) 1.5.ae
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 - 2 T + 23 T^{2}\) 1.23.ac
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 189618x do not have complex multiplication.

Modular form 189618.2.a.x

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - 2 q^{5} - q^{6} - 4 q^{7} - q^{8} + q^{9} + 2 q^{10} + q^{11} + q^{12} + 4 q^{14} - 2 q^{15} + q^{16} + q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 189618x

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
189618.n4 189618x1 \([1, 0, 1, -988147, -295745266]\) \(22106889268753393/4969545596928\) \(23987047413162442752\) \([2]\) \(6193152\) \(2.4316\) \(\Gamma_0(N)\)-optimal
189618.n2 189618x2 \([1, 0, 1, -14832627, -21987276530]\) \(74768347616680342513/5615307472896\) \(27104016647941668864\) \([2, 2]\) \(12386304\) \(2.7782\)  
189618.n3 189618x3 \([1, 0, 1, -13859187, -24997542386]\) \(-60992553706117024753/20624795251201152\) \(-99551947341654981283968\) \([2]\) \(24772608\) \(3.1247\)  
189618.n1 189618x4 \([1, 0, 1, -237317747, -1407179633650]\) \(306234591284035366263793/1727485056\) \(8338240415666304\) \([2]\) \(24772608\) \(3.1247\)