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Properties

Label 281775.bm
Number of curves $6$
Conductor $281775$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bm1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 281775.bm have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1 + T\)
\(5\)\(1\)
\(13\)\(1 + T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 281775.bm do not have complex multiplication.

Modular form 281775.2.a.bm

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 281775.bm

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
281775.bm1 281775bm6 \([1, 1, 0, -145757300, -677276009625]\) \(908031902324522977/161726530797\) \(60995082753800195203125\) \([2]\) \(37748736\) \(3.3765\)  
281775.bm2 281775bm4 \([1, 1, 0, -10035675, -8304120000]\) \(296380748763217/92608836489\) \(34927377824421175640625\) \([2, 2]\) \(18874368\) \(3.0299\)  
281775.bm3 281775bm2 \([1, 1, 0, -3930550, 2898784375]\) \(17806161424897/668584449\) \(252156301094757515625\) \([2, 2]\) \(9437184\) \(2.6833\)  
281775.bm4 281775bm1 \([1, 1, 0, -3894425, 2956476000]\) \(17319700013617/25857\) \(9751955025515625\) \([2]\) \(4718592\) \(2.3368\) \(\Gamma_0(N)\)-optimal
281775.bm5 281775bm3 \([1, 1, 0, 1596575, 10410147250]\) \(1193377118543/124806800313\) \(-47070824284754048390625\) \([2]\) \(18874368\) \(3.0299\)  
281775.bm6 281775bm5 \([1, 1, 0, 28003950, -56196007875]\) \(6439735268725823/7345472585373\) \(-2770341427610143409953125\) \([2]\) \(37748736\) \(3.3765\)