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Properties

Label 2730.bd
Number of curves $8$
Conductor $2730$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("bd1")
 
E.isogeny_class()
 

Elliptic curves in class 2730.bd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2730.bd1 2730bd7 \([1, 0, 0, -40190455, -98072518885]\) \(7179471593960193209684686321/49441793310\) \(49441793310\) \([2]\) \(110592\) \(2.5878\)  
2730.bd2 2730bd6 \([1, 0, 0, -2511905, -1532538075]\) \(1752803993935029634719121/4599740941532100\) \(4599740941532100\) \([2, 2]\) \(55296\) \(2.2412\)  
2730.bd3 2730bd8 \([1, 0, 0, -2481035, -1572033153]\) \(-1688971789881664420008241/89901485966373558750\) \(-89901485966373558750\) \([2]\) \(110592\) \(2.5878\)  
2730.bd4 2730bd4 \([1, 0, 0, -496405, -134437975]\) \(13527956825588849127121/25701087819771000\) \(25701087819771000\) \([6]\) \(36864\) \(2.0385\)  
2730.bd5 2730bd3 \([1, 0, 0, -158925, -23336703]\) \(443915739051786565201/21894701746029840\) \(21894701746029840\) \([4]\) \(27648\) \(1.8946\)  
2730.bd6 2730bd2 \([1, 0, 0, -41405, -576975]\) \(7850236389974007121/4400862921000000\) \(4400862921000000\) \([2, 6]\) \(18432\) \(1.6919\)  
2730.bd7 2730bd1 \([1, 0, 0, -25725, 1577457]\) \(1882742462388824401/11650189824000\) \(11650189824000\) \([12]\) \(9216\) \(1.3453\) \(\Gamma_0(N)\)-optimal
2730.bd8 2730bd5 \([1, 0, 0, 162715, -4536903]\) \(476437916651992691759/284661685546875000\) \(-284661685546875000\) \([6]\) \(36864\) \(2.0385\)  

Rank

sage: E.rank()
 

The elliptic curves in class 2730.bd have rank \(0\).

Complex multiplication

The elliptic curves in class 2730.bd do not have complex multiplication.

Modular form 2730.2.a.bd

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

Isogeny matrix

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.