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Properties

Label 21450cq
Number of curves $4$
Conductor $21450$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cq1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 21450cq have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(5\)\(1\)
\(11\)\(1 - T\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 21450cq do not have complex multiplication.

Modular form 21450.2.a.cq

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 21450cq

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21450.cr4 21450cq1 \([1, 0, 0, -43739213, 40767414417]\) \(592265697637387401314569/296787655248366796800\) \(4637307113255731200000000\) \([2]\) \(4055040\) \(3.4258\) \(\Gamma_0(N)\)-optimal
21450.cr2 21450cq2 \([1, 0, 0, -568027213, 5206577078417]\) \(1297212465095901089487274249/1193746061037404160000\) \(18652282203709440000000000\) \([2, 2]\) \(8110080\) \(3.7724\)  
21450.cr3 21450cq3 \([1, 0, 0, -438235213, 7649132726417]\) \(-595697118196750093952139529/1272946549598037600000000\) \(-19889789837469337500000000000\) \([2]\) \(16220160\) \(4.1189\)  
21450.cr1 21450cq4 \([1, 0, 0, -9086427213, 333377937078417]\) \(5309860874757074224246393258249/4502770931800627200\) \(70355795809384800000000\) \([2]\) \(16220160\) \(4.1189\)