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Properties

Label 8450.q
Number of curves $2$
Conductor $8450$
CM no
Rank $0$
Graph

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

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 8450.q have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(5\)\(1\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 3 T^{2}\) 1.3.a
\(7\) \( 1 + 3 T + 7 T^{2}\) 1.7.d
\(11\) \( 1 - 3 T + 11 T^{2}\) 1.11.ad
\(17\) \( 1 - 4 T + 17 T^{2}\) 1.17.ae
\(19\) \( 1 + 7 T + 19 T^{2}\) 1.19.h
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + 8 T + 29 T^{2}\) 1.29.i
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 8450.q do not have complex multiplication.

Modular form 8450.2.a.q

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

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 8450.q

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8450.q1 8450n2 \([1, -1, 1, -325015255, -2437743012753]\) \(-1762712152495281/171798691840\) \(-370061071229585981440000000\) \([]\) \(3669120\) \(3.8390\)  
8450.q2 8450n1 \([1, -1, 1, -3704005, 4472395997]\) \(-2609064081/2500000\) \(-5385097337851562500000\) \([]\) \(524160\) \(2.8660\) \(\Gamma_0(N)\)-optimal