Properties

Label 187200.x
Number of curves $4$
Conductor $187200$
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 187200.x have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 187200.x do not have complex multiplication.

Modular form 187200.2.a.x

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 187200.x

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
187200.x1 187200kv4 \([0, 0, 0, -1246010700, -16928979194000]\) \(71647584155243142409/10140000\) \(30277877760000000000\) \([2]\) \(47185920\) \(3.5910\)  
187200.x2 187200kv3 \([0, 0, 0, -89402700, -181083386000]\) \(26465989780414729/10571870144160\) \(31567435100539453440000000\) \([2]\) \(47185920\) \(3.5910\)  
187200.x3 187200kv2 \([0, 0, 0, -77882700, -264465146000]\) \(17496824387403529/6580454400\) \(19649131551129600000000\) \([2, 2]\) \(23592960\) \(3.2444\)  
187200.x4 187200kv1 \([0, 0, 0, -4154700, -5384954000]\) \(-2656166199049/2658140160\) \(-7937163987517440000000\) \([2]\) \(11796480\) \(2.8979\) \(\Gamma_0(N)\)-optimal