Properties

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

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E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 210.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
210.b1 210b7 \([1, 0, 1, -351233, -80149132]\) \(4791901410190533590281/41160000\) \(41160000\) \([2]\) \(1152\) \(1.5023\)  
210.b2 210b6 \([1, 0, 1, -21953, -1253644]\) \(1169975873419524361/108425318400\) \(108425318400\) \([2, 2]\) \(576\) \(1.1557\)  
210.b3 210b8 \([1, 0, 1, -20353, -1443724]\) \(-932348627918877961/358766164249920\) \(-358766164249920\) \([4]\) \(1152\) \(1.5023\)  
210.b4 210b4 \([1, 0, 1, -4358, -109132]\) \(9150443179640281/184570312500\) \(184570312500\) \([6]\) \(384\) \(0.95296\)  
210.b5 210b3 \([1, 0, 1, -1473, -16652]\) \(353108405631241/86318776320\) \(86318776320\) \([2]\) \(288\) \(0.80912\)  
210.b6 210b2 \([1, 0, 1, -578, 2756]\) \(21302308926361/8930250000\) \(8930250000\) \([2, 6]\) \(192\) \(0.60639\)  
210.b7 210b1 \([1, 0, 1, -498, 4228]\) \(13619385906841/6048000\) \(6048000\) \([6]\) \(96\) \(0.25981\) \(\Gamma_0(N)\)-optimal
210.b8 210b5 \([1, 0, 1, 1922, 20756]\) \(785793873833639/637994920500\) \(-637994920500\) \([12]\) \(384\) \(0.95296\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 210.2.a.b

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

Isogeny matrix

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.