LMFDB - Number field 14.0.1076631340001839899398799360.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x + 7)

gp: K = bnfinit(y^14 - 2*y + 7, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 2*x + 7);

oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - 2*x + 7)

$x^{14} - 2x + 7$ x^14 - 2*x + 7

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$14$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 7]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-1076631340001839899398799360$ -1076631340001839899398799360 $\medspace = -\,2^{15}\cdot 3\cdot 5\cdot 229\cdot 5659\cdot 1690248843237413$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$85.28$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$2^{9/4}3^{1/2}5^{1/2}229^{1/2}5659^{1/2}1690248843237413^{1/2}\approx 862235761920.1626$
Ramified primes:	$2$ , $3$ , $5$ , $229$ , $5659$ , $1690248843237413$ 2, 3, 5, 229, 5659, 1690248843237413	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-65712\!\cdots\!47290}$ )
$\Aut(K/\Q)$ :	$C_1$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$ .
This is not a CM field.

Integral basis (with respect to field generator $a$ )

$1$ , $a$ , $a^{2}$ , $a^{3}$ , $a^{4}$ , $a^{5}$ , $a^{6}$ , $a^{7}$ , $a^{8}$ , $a^{9}$ , $a^{10}$ , $a^{11}$ , $a^{12}$ , $a^{13}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Yes
Index:		$1$
Inessential primes:		None

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$6$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$-1$ -1 (order $2$ )	sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK)
Fundamental units:	$6a^{13}-7a^{12}+18a^{11}+21a^{10}-40a^{9}-22a^{8}+35a^{7}-3a^{6}+8a^{5}+51a^{4}-74a^{3}-80a^{2}+107a+29$ , $65a^{13}+162a^{12}+202a^{11}+166a^{10}+33a^{9}-158a^{8}-308a^{7}-363a^{6}-260a^{5}+11a^{4}+342a^{3}+594a^{2}+632a+263$ , $13a^{13}-10a^{12}+6a^{11}-37a^{10}+3a^{9}-53a^{8}+15a^{7}-51a^{6}+25a^{5}-49a^{4}-56a^{2}-62a-57$ , $359a^{13}+574a^{12}+520a^{11}+508a^{10}+756a^{9}+607a^{8}+374a^{7}+617a^{6}+343a^{5}-257a^{4}-40a^{3}-370a^{2}-1460a-1948$ , $16a^{13}+22a^{12}-52a^{11}+109a^{10}-184a^{9}+238a^{8}-311a^{7}+409a^{6}-448a^{5}+490a^{4}-554a^{3}+495a^{2}-415a+306$ , $81a^{13}+15a^{12}-94a^{11}-129a^{10}-28a^{9}+126a^{8}+165a^{7}+19a^{6}-169a^{5}-169a^{4}+65a^{3}+288a^{2}+190a-389$ 6a^13 - 7a^12 + 18a^11 + 21a^10 - 40a^9 - 22a^8 + 35a^7 - 3a^6 + 8a^5 + 51a^4 - 74a^3 - 80a^2 + 107a + 29, 65a^13 + 162a^12 + 202a^11 + 166a^10 + 33a^9 - 158a^8 - 308a^7 - 363a^6 - 260a^5 + 11a^4 + 342a^3 + 594a^2 + 632a + 263, 13a^13 - 10a^12 + 6a^11 - 37a^10 + 3a^9 - 53a^8 + 15a^7 - 51a^6 + 25a^5 - 49a^4 - 56a^2 - 62a - 57, 359a^13 + 574a^12 + 520a^11 + 508a^10 + 756a^9 + 607a^8 + 374a^7 + 617a^6 + 343a^5 - 257a^4 - 40a^3 - 370a^2 - 1460a - 1948, 16a^13 + 22a^12 - 52a^11 + 109a^10 - 184a^9 + 238a^8 - 311a^7 + 409a^6 - 448a^5 + 490a^4 - 554a^3 + 495a^2 - 415a + 306, 81a^13 + 15a^12 - 94a^11 - 129a^10 - 28a^9 + 126a^8 + 165a^7 + 19a^6 - 169a^5 - 169a^4 + 65a^3 + 288a^2 + 190*a - 389 (assuming GRH)	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	$327117591.886$ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 327117591.886 \cdot 1}{2\cdot\sqrt{1076631340001839899398799360}}\cr\approx \mathstrut & 1.92707880328 \end{aligned}$ (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x + 7)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^14 - 2*x + 7, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 2*x + 7);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 2*x + 7);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$S_{14}$ (as 14T63):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

A non-solvable group of order 87178291200

The 135 conjugacy class representatives for

S_{14}

Character table for

S_{14}

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and

\Q

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

Sibling fields

Degree 28 sibling:	data not computed
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	R	${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$	${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.3.0.1}{3} }$	${\href{/padicField/13.14.0.1}{14} }$	${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.3.0.1}{3} }$	${\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$	${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$	${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$	${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$	${\href{/padicField/37.14.0.1}{14} }$	${\href{/padicField/41.14.0.1}{14} }$	${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$	${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$	${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.4.0.1}{4} }$	${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of

[e_i,f_i]

for the factorization of the ideal

p\mathcal{O}_K

for

p=7

in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of

[e_i,f_i]

for the factorization of the ideal

p\mathcal{O}_K

for

p=7

in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of

[e_i,f_i]

for the factorization of the ideal

p\mathcal{O}_K

for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of

[e_i,f_i]

for the factorization of the ideal

p\mathcal{O}_K

for

p=7

in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.1.2.3a1.1	$x^{2} + 2$	$2$	$1$	$3$	$C_2$	$[3]$
	2.3.2.6a3.2	$x^{6} + 4 x^{4} + 4 x^{3} + 7 x^{2} + 6 x + 5$	$2$	$3$	$6$	$A_4\times C_2$	$[2, 2]^{6}$
	2.3.2.6a3.1	$x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 6 x + 5$	$2$	$3$	$6$	$A_4$	$[2, 2]^{3}$
$3$ 3	3.1.2.1a1.2	$x^{2} + 6$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.0a1.1	$x^{2} + 2 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	3.10.1.0a1.1	$x^{10} + 2 x^{6} + 2 x^{5} + 2 x^{4} + x + 2$	$1$	$10$	$0$	$C_{10}$	$[\ ]^{10}$
$5$ 5	$\Q_{5}$	$x + 3$	$1$	$1$	$0$	Trivial	$[\ ]$
	5.1.2.1a1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.1.0a1.1	$x^{2} + 4 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	5.9.1.0a1.1	$x^{9} + 2 x^{3} + x + 3$	$1$	$9$	$0$	$C_9$	$[\ ]^{9}$
$229$ 229	$\Q_{229}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $5$	$1$	$5$	$0$	$C_5$	$[\ ]^{5}$
		Deg $6$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
$5659$ 5659	$\Q_{5659}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $3$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
		Deg $3$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
		Deg $5$	$1$	$5$	$0$	$C_5$	$[\ ]^{5}$
$1690248843237413$ 1690248843237413	$\Q_{1690248843237413}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{1690248843237413}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $4$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
		Deg $6$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$

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Normalized defining polynomial

Invariants

Integral basis (with respect to field generator $a$ )

Class group and class number

Unit group

Class number formula

Galois group

Intermediate fields

Sibling fields

Frobenius cycle types

Local algebras for ramified primes

Spectrum of ring of integers

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	14.0.107...360.1

Degree	$14$
Signature	$[0, 7]$
Discriminant	$-1.077\times 10^{27}$
Root discriminant	$85.28$
Ramified primes	see page
Class number	$1$ (GRH)
Class group	trivial (GRH)
Galois group	$S_{14}$ (as 14T63)

Properties

Related objects

Downloads

Learn more

Integral basis (with respect to field generator aaa)

Class group and class number

Sibling fields

Local algebras for ramified primes

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Integral basis (with respect to field generator $a$ )