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Properties

Label 47040bm
Number of curves $6$
Conductor $47040$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bm1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 47040bm have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(5\)\(1 + T\)
\(7\)\(1\)
 
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
\(19\) \( 1 + 8 T + 19 T^{2}\) 1.19.i
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 47040bm do not have complex multiplication.

Modular form 47040.2.a.bm

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 47040bm

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
47040.dn4 47040bm1 \([0, -1, 0, -34365, -2440563]\) \(37256083456/525\) \(63248102400\) \([2]\) \(98304\) \(1.2127\) \(\Gamma_0(N)\)-optimal
47040.dn3 47040bm2 \([0, -1, 0, -35345, -2292975]\) \(2533446736/275625\) \(531284060160000\) \([2, 2]\) \(196608\) \(1.5593\)  
47040.dn5 47040bm3 \([0, -1, 0, 46975, -11463423]\) \(1486779836/8203125\) \(-63248102400000000\) \([2]\) \(393216\) \(1.9059\)  
47040.dn2 47040bm4 \([0, -1, 0, -133345, 16307425]\) \(34008619684/4862025\) \(37487403284889600\) \([2, 2]\) \(393216\) \(1.9059\)  
47040.dn6 47040bm5 \([0, -1, 0, 219455, 87643585]\) \(75798394558/259416045\) \(-4000322457200885760\) \([2]\) \(786432\) \(2.2524\)  
47040.dn1 47040bm6 \([0, -1, 0, -2054145, 1133828865]\) \(62161150998242/1607445\) \(24787589110824960\) \([2]\) \(786432\) \(2.2524\)