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Properties

Label 210210.fl
Number of curves $4$
Conductor $210210$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("fl1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 210210.fl 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 \(\mathbb{F}_p\)
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 210210.2.a.fl

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{8} + q^{9} + q^{10} + q^{11} + q^{12} + q^{13} + q^{15} + q^{16} + 2 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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 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 210210.fl

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
210210.fl1 210210d3 \([1, 0, 0, -27519132355, 1757112835525025]\) \(19590236683225255317943875248929/54195348396489300000\) \(6376028543498569655700000\) \([2]\) \(353894400\) \(4.4177\)  
210210.fl2 210210d4 \([1, 0, 0, -2258511235, 8842347404897]\) \(10829346205367046227129003809/5979872213745117187500000\) \(703525986074899291992187500000\) \([2]\) \(353894400\) \(4.4177\)  
210210.fl3 210210d2 \([1, 0, 0, -1720632355, 27431764225025]\) \(4788502600127122071579248929/7954695558810000000000\) \(935861977798437690000000000\) \([2, 2]\) \(176947200\) \(4.0711\)  
210210.fl4 210210d1 \([1, 0, 0, -74608675, 696060000257]\) \(-390394287570401650575649/1553162059549900800000\) \(-182727963143986279219200000\) \([4]\) \(88473600\) \(3.7245\) \(\Gamma_0(N)\)-optimal