Properties

Label 210210.fj
Number of curves $2$
Conductor $210210$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("fj1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 210210.fj have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(7\)\(1\)
\(11\)\(1 - T\)
\(13\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(17\) \( 1 + 4 T + 17 T^{2}\) 1.17.e
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 210210.fj do not have complex multiplication.

Modular form 210210.2.a.fj

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{8} + q^{9} + q^{10} + q^{11} + q^{12} + q^{13} + q^{15} + q^{16} - 4 q^{17} + q^{18} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 210210.fj

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
210210.fj1 210210b2 \([1, 0, 0, -929160, 144350100]\) \(258643243112232038887/123406016912392500\) \(42328263800950627500\) \([2]\) \(6488064\) \(2.4599\)  
210210.fj2 210210b1 \([1, 0, 0, 208340, 17177600]\) \(2915730912690561113/2058226706250000\) \(-705971760243750000\) \([2]\) \(3244032\) \(2.1133\) \(\Gamma_0(N)\)-optimal