Elliptic curve with Cremona label 28042b1 (LMFDB label 28042.a1)

• Label: 28042b1
• Conductor: $28042$
• Discriminant: $-392588$
• j-invariant: $-\frac{781229961}{392588}$
• CM: no
• Rank: $3$
• Torsion structure: trivial

Minimal Weierstrass equation

$y^2+xy=x^3-x^2-19x+49$

(, )

Mordell-Weil group structure

$\Z \oplus \Z \oplus \Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(3, 2)$	$0.51906351572454799514271430087$	$\infty$
$(-4, 9)$	$0.69290437740595095378338000160$	$\infty$
$(10, 23)$	$1.0731948609472752729464555337$	$\infty$

Integral points

$\left(-5, 3\right)$, $\left(-5, 2\right)$, $\left(-4, 9\right)$, $\left(-4, -5\right)$, $\left(-3, 10\right)$, $\left(-3, -7\right)$, $\left(0, 7\right)$, $\left(0, -7\right)$, $\left(1, 5\right)$, $\left(1, -6\right)$, $\left(2, 3\right)$, $\left(2, -5\right)$, $\left(3, 2\right)$, $\left(3, -5\right)$, $\left(4, 3\right)$, $\left(4, -7\right)$, $\left(8, 15\right)$, $\left(8, -23\right)$, $\left(10, 23\right)$, $\left(10, -33\right)$, $\left(15, 47\right)$, $\left(15, -62\right)$, $\left(17, 58\right)$, $\left(17, -75\right)$, $\left(29, 138\right)$, $\left(29, -167\right)$, $\left(52, 345\right)$, $\left(52, -397\right)$, $\left(66, 499\right)$, $\left(66, -565\right)$, $\left(88, 777\right)$, $\left(88, -865\right)$, $\left(156, 1865\right)$, $\left(156, -2021\right)$, $\left(248, 3775\right)$, $\left(248, -4023\right)$, $\left(535, 12098\right)$, $\left(535, -12633\right)$, $\left(3629, 216778\right)$, $\left(3629, -220407\right)$, $\left(4043, 255027\right)$, $\left(4043, -259070\right)$

Invariants

Conductor:	$N$	=	$28042$	=	$2 \cdot 7 \cdot 2003$
Discriminant:	$\Delta$	=	$-392588$	=	$-1 \cdot 2^{2} \cdot 7^{2} \cdot 2003$
j-invariant:	$j$	=	$-\frac{781229961}{392588}$	=	$-1 \cdot 2^{-2} \cdot 3^{3} \cdot 7^{-2} \cdot 307^{3} \cdot 2003^{-1}$
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:	$h_{\mathrm{Faltings}}$	≈	$-0.22166656025028913309077588733$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-0.22166656025028913309077588733$
$abc$ quality:	$Q$	≈	$0.7320841350295747$
Szpiro ratio:	$\sigma_{m}$	≈	$2.060394328758852$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$3$
Mordell-Weil rank:	$r$	=	$3$
Regulator:	$\mathrm{Reg}(E/\Q)$	≈	$0.37085236964942813880449256642$
Real period:	$\Omega$	≈	$2.7968572408968871454006184953$
Tamagawa product:	$\prod_{p}c_p$	=	$4$  = $2\cdot2\cdot1$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$
Special value:	$L^{(3)}(E,1)/3!$	≈	$4.1488845414310883002200022061$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	≈	$1$    (rounded)

BSD formula

$\displaystyle 4.148884541 \approx L^{(3)}(E,1)/3! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 2.796857 \cdot 0.370852 \cdot 4}{1^2} \approx 4.148884541$

Modular invariants

Modular form   28042.2.a.a

$q - q^{2} - 3 q^{3} + q^{4} - 4 q^{5} + 3 q^{6} - q^{7} - q^{8} + 6 q^{9} + 4 q^{10} - 6 q^{11} - 3 q^{12} - 3 q^{13} + q^{14} + 12 q^{15} + q^{16} - 6 q^{17} - 6 q^{18} - 4 q^{19} + O(q^{20})$

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

Modular degree:	16416
$\Gamma_0(N)$-optimal:	yes
Manin constant:	1

Local data at primes of bad reduction

This elliptic curve is semistable. There are 3 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$2$	$I_{2}$	nonsplit multiplicative	1	1	2	2
$7$	$2$	$I_{2}$	nonsplit multiplicative	1	1	2	2
$2003$	$1$	$I_{1}$	nonsplit multiplicative	1	1	1	1

Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$.

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $4006 = 2 \cdot 2003$, index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 4005 & 0 \end{array}\right),\left(\begin{array}{rr} 4005 & 2 \\ 4004 & 3 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 5 & 3 \end{array}\right)$.

The torsion field $K:=\Q(E[4006])$ is a degree-$48264528456144$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4006\Z)$.

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
$2$	nonsplit multiplicative	$4$	$2003$
$7$	nonsplit multiplicative	$8$	$4006 = 2 \cdot 2003$
$2003$	nonsplit multiplicative	$2004$	$14 = 2 \cdot 7$

Isogenies

This curve has no rational isogenies. Its isogeny class 28042b consists of this curve only.

Twists

This elliptic curve is its own minimal quadratic twist.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$3$	3.1.2003.1	$\Z/2\Z$	not in database
$6$	6.0.8036054027.1	$\Z/2\Z \oplus \Z/2\Z$	not in database
$8$	deg 8	$\Z/3\Z$	not in database
$12$	deg 12	$\Z/4\Z$	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	2003
Reduction type	nonsplit	ss	ord	nonsplit	ord	ord	ord	ord	ord	ord	ss	ord	ord	ord	ord	nonsplit
$\lambda$-invariant(s)	5	3,3	3	3	3	3	5	3	3	3	3,3	3	3	3	3	?
$\mu$-invariant(s)	0	0,0	0	0	0	0	0	0	0	0	0,0	0	0	0	0	?

An entry ? indicates that the invariants have not yet been computed.

$p$-adic regulators

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.