Elliptic curve with LMFDB label 2574.u3 (Cremona label 2574x2)

• Label: 2574.u3
• Conductor: $2574$
• Discriminant: $1.839\times 10^{14}$
• j-invariant: $\frac{645532578015625}{252306960048}$
• CM: no
• Rank: $1$
• Torsion structure: $\Z/{2}\Z$

Minimal Weierstrass equation

$y^2+xy+y=x^3-x^2-16205x-448315$

(, )

Mordell-Weil group structure

$\Z \oplus \Z/{2}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(-39, 370)$	$0.20568461723680442357928171822$	$\infty$
$(-117/4, 113/8)$	$0$	$2$

Integral points

$\left(-65, 604\right)$, $\left(-65, -540\right)$, $\left(-39, 370\right)$, $\left(-39, -332\right)$, $\left(177, 1396\right)$, $\left(177, -1574\right)$, $\left(351, 5908\right)$, $\left(351, -6260\right)$, $\left(1365, 49510\right)$, $\left(1365, -50876\right)$

Invariants

Conductor:	$N$	=	$2574$	=	$2 \cdot 3^{2} \cdot 11 \cdot 13$
Discriminant:	$\Delta$	=	$183931773874992$	=	$2^{4} \cdot 3^{9} \cdot 11^{2} \cdot 13^{6}$
j-invariant:	$j$	=	$\frac{645532578015625}{252306960048}$	=	$2^{-4} \cdot 3^{-3} \cdot 5^{6} \cdot 11^{-2} \cdot 13^{-6} \cdot 3457^{3}$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z$    (no potential complex multiplication)
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{SU}(2)$
Faltings height:	$h_{\mathrm{Faltings}}$	≈	$1.4358247187422841881905294782$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$0.88651857440822934249290685974$
$abc$ quality:	$Q$	≈	$1.033360901289125$
Szpiro ratio:	$\sigma_{m}$	≈	$5.181669333312839$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$1$
Mordell-Weil rank:	$r$	=	$1$
Regulator:	$\mathrm{Reg}(E/\Q)$	≈	$0.20568461723680442357928171822$
Real period:	$\Omega$	≈	$0.43749823068009232121298983373$
Tamagawa product:	$\prod_{p}c_p$	=	$192$  = $2^{2}\cdot2^{2}\cdot2\cdot( 2 \cdot 3 )$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$2$
Special value:	$L'(E,1)$	≈	$4.3193594937222698374315044419$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	≈	$1$    (rounded)

BSD formula

$\begin{aligned} 4.319359494 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.437498 \cdot 0.205685 \cdot 192}{2^2} \\ & \approx 4.319359494\end{aligned}$

Modular invariants

Modular form   2574.2.a.u

$q + q^{2} + q^{4} - 4 q^{7} + q^{8} + q^{11} + q^{13} - 4 q^{14} + q^{16} - 4 q^{19} + O(q^{20})$

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

Modular degree:	9216
$\Gamma_0(N)$-optimal:	no
Manin constant:	1

Local data at primes of bad reduction

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

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$4$	$I_{4}$	split multiplicative	-1	1	4	4
$3$	$4$	$I_{3}^{*}$	additive	-1	2	9	3
$11$	$2$	$I_{2}$	split multiplicative	-1	1	2	2
$13$	$6$	$I_{6}$	split multiplicative	-1	1	6	6

Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$	mod-$\ell$ image	$\ell$-adic image
$2$	2B	2.3.0.1
$3$	3B.1.2	3.8.0.2

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13$, index $96$, genus $1$, and generators

$\left(\begin{array}{rr} 1705 & 12 \\ 1704 & 13 \end{array}\right),\left(\begin{array}{rr} 1002 & 1585 \\ 1001 & 1574 \end{array}\right),\left(\begin{array}{rr} 937 & 12 \\ 474 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 925 & 12 \\ 402 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1138 & 1705 \\ 1155 & 20 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 1666 & 1707 \end{array}\right)$.

The torsion field $K:=\Q(E[1716])$ is a degree-$16605388800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1716\Z)$.

The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.

$\ell$	Reduction type	Serre weight	Serre conductor
$2$	split multiplicative	$4$	$9 = 3^{2}$
$3$	additive	$2$	$22 = 2 \cdot 11$
$11$	split multiplicative	$12$	$234 = 2 \cdot 3^{2} \cdot 13$
$13$	split multiplicative	$14$	$198 = 2 \cdot 3^{2} \cdot 11$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 2574.u consists of 4 curves linked by isogenies of degrees dividing 6.

Twists

The minimal quadratic twist of this elliptic curve is 858.c3, its twist by $-3$.

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
$2$	$\Q(\sqrt{3})$	$\Z/2\Z \oplus \Z/2\Z$	not in database
$2$	$\Q(\sqrt{-3})$	$\Z/6\Z$	not in database
$3$	3.1.1452.1	$\Z/6\Z$	not in database
$4$	4.0.245388.1	$\Z/4\Z$	not in database
$4$	$\Q(\zeta_{12})$	$\Z/2\Z \oplus \Z/6\Z$	not in database
$6$	6.0.6324912.1	$\Z/3\Z \oplus \Z/6\Z$	not in database
$6$	6.2.101198592.1	$\Z/2\Z \oplus \Z/6\Z$	not in database
$8$	8.4.976966189056.8	$\Z/2\Z \oplus \Z/4\Z$	not in database
$8$	8.0.8670998958336.15	$\Z/2\Z \oplus \Z/4\Z$	not in database
$8$	8.0.541937434896.3	$\Z/12\Z$	not in database
$12$	12.0.10241155022782464.1	$\Z/6\Z \oplus \Z/6\Z$	not in database
$12$	deg 12	$\Z/12\Z$	not in database
$16$	deg 16	$\Z/8\Z$	not in database
$16$	deg 16	$\Z/2\Z \oplus \Z/12\Z$	not in database
$16$	deg 16	$\Z/2\Z \oplus \Z/12\Z$	not in database
$18$	18.0.7107274348572364782671678030769311232768.1	$\Z/18\Z$	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.

Iwasawa invariants

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

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

$p$-adic regulators

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