Elliptic curve with LMFDB label 47040.u4 (Cremona label 47040ee5)

• Label: 47040.u4
• Conductor: $47040$
• Discriminant: $1.041\times 10^{15}$
• j-invariant: \( \frac{2656166199049}{33750} \)
• CM: no
• Rank: $1$
• Torsion structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2=x^3-x^2-904801x+331564801\)

Mordell-Weil group structure

\(\Z\times \Z/{2}\Z\)

Infinite order Mordell-Weil generator and height

\(P\)	=	\( \left(1176, 29875\right) \)
\(\hat{h}(P)\)	≈	$5.1802012412850831819271429283$

Torsion generators

\( \left(551, 0\right) \)

Integral points

\( \left(551, 0\right) \), \((1176,\pm 29875)\)

Invariants

Conductor:	\( 47040 \)	=	\(2^{6} \cdot 3 \cdot 5 \cdot 7^{2}\)
Discriminant:	\(1040883056640000 \)	=	\(2^{19} \cdot 3^{3} \cdot 5^{4} \cdot 7^{6} \)
j-invariant:	\( \frac{2656166199049}{33750} \)	=	\(2^{-1} \cdot 3^{-3} \cdot 5^{-4} \cdot 11^{3} \cdot 1259^{3}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$

BSD invariants

Analytic rank:	\(1\)
Regulator:	\(5.1802012412850831819271429283\)
Real period:	\(0.44792292729553569385475488502\)
Tamagawa product:	\( 8 \)  = \( 2\cdot1\cdot2\cdot2 \)
Torsion order:	\(2\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 47040.2.a.u

\( q - q^{3} - q^{5} + q^{9} + 2q^{13} + q^{15} - 6q^{17} + 4q^{19} + O(q^{20}) \)

For more coefficients, see the Downloads section to the right.

Modular degree:	442368
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L'(E,1) \) ≈ \( 4.6406618079527641368754578926211396871 \)

Local data

This elliptic curve is not semistable. There are 4 primes of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(2\)	\(I_9^{*}\)	Additive	-1	6	19	1
\(3\)	\(1\)	\(I_{3}\)	Non-split multiplicative	1	1	3	3
\(5\)	\(2\)	\(I_{4}\)	Non-split multiplicative	1	1	4	4
\(7\)	\(2\)	\(I_0^{*}\)	Additive	-1	2	6	0

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X13.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 6.

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime	Image of Galois representation
\(2\)	B
\(3\)	B

$p$-adic data

$p$-adic regulators

\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	add	nonsplit	nonsplit	add	ss	ordinary	ordinary	ordinary	ss	ordinary	ordinary	ordinary	ordinary	ordinary	ss
$\lambda$-invariant(s)	-	1	1	-	1,5	1	1	1	3,1	1	1	1	1	1	1,1
$\mu$-invariant(s)	-	0	0	-	0,0	0	0	0	0,0	0	0	0	0	0	0,0

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 47040.u consists of 6 curves linked by isogenies of degrees dividing 12.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	\(\Q(\sqrt{6}) \)	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$2$	\(\Q(\sqrt{7}) \)	\(\Z/4\Z\)	Not in database
$2$	\(\Q(\sqrt{42}) \)	\(\Z/4\Z\)	Not in database
$2$	\(\Q(\sqrt{14}) \)	\(\Z/6\Z\)	Not in database
$4$	\(\Q(\sqrt{6}, \sqrt{7})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$4$	\(\Q(\sqrt{6}, \sqrt{14})\)	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$4$	\(\Q(\sqrt{2}, \sqrt{7})\)	\(\Z/12\Z\)	Not in database
$4$	\(\Q(\sqrt{3}, \sqrt{14})\)	\(\Z/12\Z\)	Not in database
$6$	6.0.11854080000.11	\(\Z/6\Z\)	Not in database
$8$	8.0.7341411926016.66	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	8.8.226586787840000.6	\(\Z/8\Z\)	Not in database
$8$	8.0.17923368960000.35	\(\Z/8\Z\)	Not in database
$8$	8.8.12745506816.1	\(\Z/2\Z \times \Z/12\Z\)	Not in database
$12$	Deg 12	\(\Z/3\Z \times \Z/6\Z\)	Not in database
$12$	Deg 12	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$12$	Deg 12	\(\Z/12\Z\)	Not in database
$12$	Deg 12	\(\Z/12\Z\)	Not in database
$16$	Deg 16	\(\Z/4\Z \times \Z/4\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/8\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/8\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/12\Z\)	Not in database
$16$	Deg 16	\(\Z/24\Z\)	Not in database
$16$	Deg 16	\(\Z/24\Z\)	Not in database
$18$	18.6.6969632870593768497248364134400000000.1	\(\Z/18\Z\)	Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.