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Properties

Label 450g
Number of curves $8$
Conductor $450$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("g1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 450g have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(11\) \( 1 - 3 T + 11 T^{2}\) 1.11.ad
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 - 3 T + 17 T^{2}\) 1.17.ad
\(19\) \( 1 - 5 T + 19 T^{2}\) 1.19.af
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 450g do not have complex multiplication.

Modular form 450.2.a.g

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + 4 q^{7} - q^{8} - 2 q^{13} - 4 q^{14} + q^{16} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 450g

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450.d8 450g1 \([1, -1, 0, 333, -7259]\) \(357911/2160\) \(-24603750000\) \([2]\) \(384\) \(0.67574\) \(\Gamma_0(N)\)-optimal
450.d6 450g2 \([1, -1, 0, -4167, -92759]\) \(702595369/72900\) \(830376562500\) \([2, 2]\) \(768\) \(1.0223\)  
450.d7 450g3 \([1, -1, 0, -3042, 212116]\) \(-273359449/1536000\) \(-17496000000000\) \([2]\) \(1152\) \(1.2250\)  
450.d4 450g4 \([1, -1, 0, -64917, -6350009]\) \(2656166199049/33750\) \(384433593750\) \([2]\) \(1536\) \(1.3689\)  
450.d5 450g5 \([1, -1, 0, -15417, 638491]\) \(35578826569/5314410\) \(60534451406250\) \([2]\) \(1536\) \(1.3689\)  
450.d3 450g6 \([1, -1, 0, -75042, 7916116]\) \(4102915888729/9000000\) \(102515625000000\) \([2, 2]\) \(2304\) \(1.5716\)  
450.d2 450g7 \([1, -1, 0, -102042, 1733116]\) \(10316097499609/5859375000\) \(66741943359375000\) \([2]\) \(4608\) \(1.9182\)  
450.d1 450g8 \([1, -1, 0, -1200042, 506291116]\) \(16778985534208729/81000\) \(922640625000\) \([2]\) \(4608\) \(1.9182\)