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Properties

Label 4800bp
Number of curves $8$
Conductor $4800$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bp1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 4800bp have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 4800bp do not have complex multiplication.

Modular form 4800.2.a.bp

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - 4 q^{7} + q^{9} + 2 q^{13} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 4800bp

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4800.d8 4800bp1 \([0, -1, 0, 2367, -136863]\) \(357911/2160\) \(-8847360000000\) \([2]\) \(9216\) \(1.1662\) \(\Gamma_0(N)\)-optimal
4800.d6 4800bp2 \([0, -1, 0, -29633, -1768863]\) \(702595369/72900\) \(298598400000000\) \([2, 2]\) \(18432\) \(1.5127\)  
4800.d7 4800bp3 \([0, -1, 0, -21633, 4015137]\) \(-273359449/1536000\) \(-6291456000000000\) \([2]\) \(27648\) \(1.7155\)  
4800.d4 4800bp4 \([0, -1, 0, -461633, -120568863]\) \(2656166199049/33750\) \(138240000000000\) \([2]\) \(36864\) \(1.8593\)  
4800.d5 4800bp5 \([0, -1, 0, -109633, 12071137]\) \(35578826569/5314410\) \(21767823360000000\) \([2]\) \(36864\) \(1.8593\)  
4800.d3 4800bp6 \([0, -1, 0, -533633, 149935137]\) \(4102915888729/9000000\) \(36864000000000000\) \([2, 2]\) \(55296\) \(2.0620\)  
4800.d2 4800bp7 \([0, -1, 0, -725633, 32623137]\) \(10316097499609/5859375000\) \(24000000000000000000\) \([2]\) \(110592\) \(2.4086\)  
4800.d1 4800bp8 \([0, -1, 0, -8533633, 9597935137]\) \(16778985534208729/81000\) \(331776000000000\) \([2]\) \(110592\) \(2.4086\)