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Properties

Label 330.d
Number of curves $6$
Conductor $330$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("d1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 330.d have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(5\)\(1 - T\)
\(11\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 330.d do not have complex multiplication.

Modular form 330.2.a.d

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + q^{10} + q^{11} - q^{12} + 6 q^{13} - q^{15} + q^{16} + 2 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 330.d

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
330.d1 330c5 \([1, 1, 1, -171085, -27308713]\) \(553808571467029327441/12529687500\) \(12529687500\) \([2]\) \(1536\) \(1.4618\)  
330.d2 330c4 \([1, 1, 1, -11825, 488927]\) \(182864522286982801/463015182960\) \(463015182960\) \([4]\) \(768\) \(1.1152\)  
330.d3 330c3 \([1, 1, 1, -10705, -429025]\) \(135670761487282321/643043610000\) \(643043610000\) \([2, 2]\) \(768\) \(1.1152\)  
330.d4 330c6 \([1, 1, 1, -5205, -862425]\) \(-15595206456730321/310672490129100\) \(-310672490129100\) \([2]\) \(1536\) \(1.4618\)  
330.d5 330c2 \([1, 1, 1, -1025, 767]\) \(119102750067601/68309049600\) \(68309049600\) \([2, 4]\) \(384\) \(0.76861\)  
330.d6 330c1 \([1, 1, 1, 255, 255]\) \(1833318007919/1070530560\) \(-1070530560\) \([4]\) \(192\) \(0.42204\) \(\Gamma_0(N)\)-optimal