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Properties

Label 201810.bp
Number of curves $8$
Conductor $201810$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bp1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 201810.bp have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(5\)\(1 + T\)
\(7\)\(1 - T\)
\(31\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 201810.bp do not have complex multiplication.

Modular form 201810.2.a.bp

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 201810.bp

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
201810.bp1 201810be8 \([1, 1, 1, -6199431, -3740417697]\) \(29689921233686449/10380965400750\) \(9213145005499265160750\) \([2]\) \(17418240\) \(2.9160\)  
201810.bp2 201810be5 \([1, 1, 1, -5536341, -5016283581]\) \(21145699168383889/2593080\) \(2301368045127480\) \([2]\) \(5806080\) \(2.3667\)  
201810.bp3 201810be6 \([1, 1, 1, -2595681, 1565743803]\) \(2179252305146449/66177562500\) \(58732830318357562500\) \([2, 2]\) \(8709120\) \(2.5694\)  
201810.bp4 201810be3 \([1, 1, 1, -2576461, 1590706739]\) \(2131200347946769/2058000\) \(1826482575498000\) \([2]\) \(4354560\) \(2.2229\)  
201810.bp5 201810be2 \([1, 1, 1, -346941, -78050541]\) \(5203798902289/57153600\) \(50724030382401600\) \([2, 2]\) \(2903040\) \(2.0201\)  
201810.bp6 201810be4 \([1, 1, 1, -77861, -195692317]\) \(-58818484369/18600435000\) \(-16507954530701235000\) \([2]\) \(5806080\) \(2.3667\)  
201810.bp7 201810be1 \([1, 1, 1, -39421, 1043603]\) \(7633736209/3870720\) \(3435278248120320\) \([2]\) \(1451520\) \(1.6736\) \(\Gamma_0(N)\)-optimal
201810.bp8 201810be7 \([1, 1, 1, 700549, 5274661799]\) \(42841933504271/13565917968750\) \(-12039802133409667968750\) \([2]\) \(17418240\) \(2.9160\)