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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("hd1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 47040hd 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.c
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 47040.2.a.hd

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 47040hd

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
47040.he4 47040hd1 \([0, 1, 0, -144125, 21011955]\) \(2748251600896/2205\) \(265642030080\) \([2]\) \(196608\) \(1.4976\) \(\Gamma_0(N)\)-optimal
47040.he3 47040hd2 \([0, 1, 0, -145105, 20710703]\) \(175293437776/4862025\) \(9371850821222400\) \([2, 2]\) \(393216\) \(1.8441\)  
47040.he5 47040hd3 \([0, 1, 0, 31295, 68021183]\) \(439608956/259416045\) \(-2000161228600442880\) \([2]\) \(786432\) \(2.1907\)  
47040.he2 47040hd4 \([0, 1, 0, -337185, -45864225]\) \(549871953124/200930625\) \(1549224319426560000\) \([2, 2]\) \(786432\) \(2.1907\)  
47040.he6 47040hd5 \([0, 1, 0, 1034815, -323282625]\) \(7947184069438/7533176175\) \(-116165265825801830400\) \([4]\) \(1572864\) \(2.5373\)  
47040.he1 47040hd6 \([0, 1, 0, -4782465, -4026167937]\) \(784478485879202/221484375\) \(3415397529600000000\) \([2]\) \(1572864\) \(2.5373\)