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Properties

Label 330330.bc
Number of curves $6$
Conductor $330330$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bc1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 330330.bc have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(5\)\(1 - T\)
\(7\)\(1 + T\)
\(11\)\(1\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 330330.bc do not have complex multiplication.

Modular form 330330.2.a.bc

Copy content 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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 330330.bc

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
330330.bc1 330330bc5 \([1, 1, 0, -55951612, 160900594936]\) \(10934663514379917006241/12996826171875000\) \(23024670369873046875000\) \([2]\) \(62914560\) \(3.2022\)  
330330.bc2 330330bc4 \([1, 1, 0, -40274852, -98395014384]\) \(4078208988807294650401/359723582400\) \(637272269360126400\) \([2]\) \(31457280\) \(2.8556\)  
330330.bc3 330330bc3 \([1, 1, 0, -4420132, 1080862864]\) \(5391051390768345121/2833965225000000\) \(5020542267966225000000\) \([2, 2]\) \(31457280\) \(2.8556\)  
330330.bc4 330330bc2 \([1, 1, 0, -2522852, -1530932784]\) \(1002404925316922401/9348917760000\) \(16562178095823360000\) \([2, 2]\) \(15728640\) \(2.5091\)  
330330.bc5 330330bc1 \([1, 1, 0, -44772, -57466416]\) \(-5602762882081/801531494400\) \(-1419961935750758400\) \([2]\) \(7864320\) \(2.1625\) \(\Gamma_0(N)\)-optimal
330330.bc6 330330bc6 \([1, 1, 0, 16754868, 8453997864]\) \(293623352309352854879/187320324116835000\) \(-331849380712744329435000\) \([2]\) \(62914560\) \(3.2022\)