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Properties

Label 6720bw
Number of curves $8$
Conductor $6720$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bw1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 6720bw have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 6720bw do not have complex multiplication.

Modular form 6720.2.a.bw

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} - q^{5} - q^{7} + q^{9} - 2 q^{13} - q^{15} - 6 q^{17} + 8 q^{19} + O(q^{20})\) 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 6720bw

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6720.bi7 6720bw1 \([0, 1, 0, -31841, -2196705]\) \(13619385906841/6048000\) \(1585446912000\) \([2]\) \(18432\) \(1.2995\) \(\Gamma_0(N)\)-optimal
6720.bi6 6720bw2 \([0, 1, 0, -36961, -1448161]\) \(21302308926361/8930250000\) \(2341011456000000\) \([2, 2]\) \(36864\) \(1.6461\)  
6720.bi5 6720bw3 \([0, 1, 0, -94241, 8431455]\) \(353108405631241/86318776320\) \(22627949299630080\) \([2]\) \(55296\) \(1.8488\)  
6720.bi4 6720bw4 \([0, 1, 0, -278881, 55596575]\) \(9150443179640281/184570312500\) \(48384000000000000\) \([2]\) \(73728\) \(1.9927\)  
6720.bi8 6720bw5 \([0, 1, 0, 123039, -10504161]\) \(785793873833639/637994920500\) \(-167246540439552000\) \([2]\) \(73728\) \(1.9927\)  
6720.bi2 6720bw6 \([0, 1, 0, -1404961, 640460639]\) \(1169975873419524361/108425318400\) \(28423046666649600\) \([2, 2]\) \(110592\) \(2.1954\)  
6720.bi1 6720bw7 \([0, 1, 0, -22478881, 41013876575]\) \(4791901410190533590281/41160000\) \(10789847040000\) \([2]\) \(221184\) \(2.5420\)  
6720.bi3 6720bw8 \([0, 1, 0, -1302561, 737883999]\) \(-932348627918877961/358766164249920\) \(-94048397361131028480\) \([2]\) \(221184\) \(2.5420\)