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Properties

Label 30.a
Number of curves $8$
Conductor $30$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("a1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 30.a have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 30.a do not have complex multiplication.

Modular form 30.2.a.a

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 30.a

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
30.a1 30a7 \([1, 0, 1, -5334, -150368]\) \(16778985534208729/81000\) \(81000\) \([2]\) \(24\) \(0.56417\)  
30.a2 30a8 \([1, 0, 1, -454, -544]\) \(10316097499609/5859375000\) \(5859375000\) \([2]\) \(24\) \(0.56417\)  
30.a3 30a6 \([1, 0, 1, -334, -2368]\) \(4102915888729/9000000\) \(9000000\) \([2, 2]\) \(12\) \(0.21759\)  
30.a4 30a5 \([1, 0, 1, -289, 1862]\) \(2656166199049/33750\) \(33750\) \([6]\) \(8\) \(0.014860\)  
30.a5 30a4 \([1, 0, 1, -69, -194]\) \(35578826569/5314410\) \(5314410\) \([6]\) \(8\) \(0.014860\)  
30.a6 30a2 \([1, 0, 1, -19, 26]\) \(702595369/72900\) \(72900\) \([2, 6]\) \(4\) \(-0.33171\)  
30.a7 30a3 \([1, 0, 1, -14, -64]\) \(-273359449/1536000\) \(-1536000\) \([2]\) \(6\) \(-0.12898\)  
30.a8 30a1 \([1, 0, 1, 1, 2]\) \(357911/2160\) \(-2160\) \([6]\) \(2\) \(-0.67829\) \(\Gamma_0(N)\)-optimal