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Properties

Label 210e
Number of curves $8$
Conductor $210$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("e1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 210e have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(7\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 210e do not have complex multiplication.

Modular form 210.2.a.e

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - q^{7} + q^{8} + q^{9} + q^{10} - 4 q^{11} + q^{12} - 2 q^{13} - q^{14} + q^{15} + q^{16} + 2 q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 210e

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
210.e7 210e1 \([1, 0, 0, 210, 900]\) \(1023887723039/928972800\) \(-928972800\) \([8]\) \(128\) \(0.40813\) \(\Gamma_0(N)\)-optimal
210.e6 210e2 \([1, 0, 0, -1070, 7812]\) \(135487869158881/51438240000\) \(51438240000\) \([2, 8]\) \(256\) \(0.75471\)  
210.e5 210e3 \([1, 0, 0, -7550, -247500]\) \(47595748626367201/1215506250000\) \(1215506250000\) \([2, 4]\) \(512\) \(1.1013\)  
210.e4 210e4 \([1, 0, 0, -15070, 710612]\) \(378499465220294881/120530818800\) \(120530818800\) \([8]\) \(512\) \(1.1013\)  
210.e2 210e5 \([1, 0, 0, -120050, -16020000]\) \(191342053882402567201/129708022500\) \(129708022500\) \([2, 2]\) \(1024\) \(1.4479\)  
210.e8 210e6 \([1, 0, 0, 1270, -789048]\) \(226523624554079/269165039062500\) \(-269165039062500\) \([4]\) \(1024\) \(1.4479\)  
210.e1 210e7 \([1, 0, 0, -1920800, -1024800150]\) \(783736670177727068275201/360150\) \(360150\) \([2]\) \(2048\) \(1.7944\)  
210.e3 210e8 \([1, 0, 0, -119300, -16229850]\) \(-187778242790732059201/4984939585440150\) \(-4984939585440150\) \([2]\) \(2048\) \(1.7944\)