LMFDB - Number field 14.2.391066968923137.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 5*x^12 + 4*x^11 - 14*x^10 + 5*x^9 + 13*x^8 - 9*x^7 - 9*x^6 + 8*x^5 + 4*x^4 - 5*x^3 + x - 1)

gp: K = bnfinit(y^14 - 4*y^13 + 5*y^12 + 4*y^11 - 14*y^10 + 5*y^9 + 13*y^8 - 9*y^7 - 9*y^6 + 8*y^5 + 4*y^4 - 5*y^3 + y - 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 4*x^13 + 5*x^12 + 4*x^11 - 14*x^10 + 5*x^9 + 13*x^8 - 9*x^7 - 9*x^6 + 8*x^5 + 4*x^4 - 5*x^3 + x - 1);

oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - 4*x^13 + 5*x^12 + 4*x^11 - 14*x^10 + 5*x^9 + 13*x^8 - 9*x^7 - 9*x^6 + 8*x^5 + 4*x^4 - 5*x^3 + x - 1)

$x^{14} - 4 x^{13} + 5 x^{12} + 4 x^{11} - 14 x^{10} + 5 x^{9} + 13 x^{8} - 9 x^{7} - 9 x^{6} + 8 x^{5} + \cdots - 1$ x^14 - 4*x^13 + 5*x^12 + 4*x^11 - 14*x^10 + 5*x^9 + 13*x^8 - 9*x^7 - 9*x^6 + 8*x^5 + 4*x^4 - 5*x^3 + x - 1

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:	$[2, 6]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$391066968923137$ 391066968923137 $\medspace = 13^{4}\cdot 97\cdot 109^{4}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$11.02$	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:	$13^{1/2}97^{1/2}109^{1/2}\approx 370.7411495909242$
Ramified primes:	$13$ , $97$ , $109$ 13, 97, 109	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{97})$
$\Aut(K/\Q)$ :	$C_2$	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 $a$ )

$1$ , $a$ , $a^{2}$ , $a^{3}$ , $a^{4}$ , $a^{5}$ , $a^{6}$ , $a^{7}$ , $a^{8}$ , $a^{9}$ , $a^{10}$ , $a^{11}$ , $a^{12}$ , $\frac{1}{367}a^{13}-\frac{152}{367}a^{12}+\frac{114}{367}a^{11}+\frac{14}{367}a^{10}+\frac{116}{367}a^{9}+\frac{86}{367}a^{8}+\frac{130}{367}a^{7}-\frac{165}{367}a^{6}-\frac{178}{367}a^{5}-\frac{72}{367}a^{4}+\frac{17}{367}a^{3}+\frac{48}{367}a^{2}-\frac{131}{367}a-\frac{62}{367}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, 1/367*a^13 - 152/367*a^12 + 114/367*a^11 + 14/367*a^10 + 116/367*a^9 + 86/367*a^8 + 130/367*a^7 - 165/367*a^6 - 178/367*a^5 - 72/367*a^4 + 17/367*a^3 + 48/367*a^2 - 131/367*a - 62/367

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

Ideal class group:		Trivial group, which has order $1$ sage: K.class_group().invariants() gp: K.clgp magma: ClassGroup(K); oscar: class_group(K)
Narrow class group:		Trivial group, which has order $1$

Unit group

sage: UK = K.unit_group()

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

oscar: UK, fUK = unit_group(OK)

Rank:	$7$	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:	$a$ , $\frac{185}{367}a^{13}-\frac{595}{367}a^{12}+\frac{538}{367}a^{11}+\frac{755}{367}a^{10}-\frac{1294}{367}a^{9}-\frac{238}{367}a^{8}+\frac{1296}{367}a^{7}-\frac{64}{367}a^{6}-\frac{1368}{367}a^{5}+\frac{626}{367}a^{4}+\frac{209}{367}a^{3}-\frac{662}{367}a^{2}-\frac{13}{367}a+\frac{274}{367}$ , $\frac{38}{367}a^{13}-\frac{271}{367}a^{12}+\frac{662}{367}a^{11}-\frac{569}{367}a^{10}-\frac{363}{367}a^{9}+\frac{699}{367}a^{8}+\frac{169}{367}a^{7}-\frac{398}{367}a^{6}-\frac{525}{367}a^{5}+\frac{567}{367}a^{4}-\frac{88}{367}a^{3}-\frac{11}{367}a^{2}-\frac{207}{367}a-\frac{154}{367}$ , $\frac{35}{367}a^{13}-\frac{182}{367}a^{12}+\frac{320}{367}a^{11}+\frac{123}{367}a^{10}-\frac{1078}{367}a^{9}+\frac{808}{367}a^{8}+\frac{1247}{367}a^{7}-\frac{1738}{367}a^{6}-\frac{725}{367}a^{5}+\frac{1517}{367}a^{4}+\frac{228}{367}a^{3}-\frac{889}{367}a^{2}-\frac{181}{367}a+\frac{32}{367}$ , $\frac{32}{367}a^{13}-\frac{93}{367}a^{12}-\frac{22}{367}a^{11}+\frac{448}{367}a^{10}-\frac{692}{367}a^{9}+\frac{183}{367}a^{8}+\frac{490}{367}a^{7}-\frac{876}{367}a^{6}+\frac{176}{367}a^{5}+\frac{632}{367}a^{4}-\frac{190}{367}a^{3}-\frac{666}{367}a^{2}+\frac{212}{367}a+\frac{218}{367}$ , $\frac{143}{367}a^{13}-\frac{817}{367}a^{12}+\frac{1622}{367}a^{11}-\frac{567}{367}a^{10}-\frac{2496}{367}a^{9}+\frac{2756}{367}a^{8}+\frac{974}{367}a^{7}-\frac{2309}{367}a^{6}-\frac{498}{367}a^{5}+\frac{1448}{367}a^{4}+\frac{229}{367}a^{3}-\frac{476}{367}a^{2}-\frac{16}{367}a-\frac{58}{367}$ , $\frac{33}{367}a^{13}-\frac{245}{367}a^{12}+\frac{459}{367}a^{11}+\frac{95}{367}a^{10}-\frac{1310}{367}a^{9}+\frac{636}{367}a^{8}+\frac{1721}{367}a^{7}-\frac{1408}{367}a^{6}-\frac{1470}{367}a^{5}+\frac{1294}{367}a^{4}+\frac{1295}{367}a^{3}-\frac{618}{367}a^{2}-\frac{653}{367}a+\frac{156}{367}$ a, 185/367a^13 - 595/367a^12 + 538/367a^11 + 755/367a^10 - 1294/367a^9 - 238/367a^8 + 1296/367a^7 - 64/367a^6 - 1368/367a^5 + 626/367a^4 + 209/367a^3 - 662/367a^2 - 13/367a + 274/367, 38/367a^13 - 271/367a^12 + 662/367a^11 - 569/367a^10 - 363/367a^9 + 699/367a^8 + 169/367a^7 - 398/367a^6 - 525/367a^5 + 567/367a^4 - 88/367a^3 - 11/367a^2 - 207/367a - 154/367, 35/367a^13 - 182/367a^12 + 320/367a^11 + 123/367a^10 - 1078/367a^9 + 808/367a^8 + 1247/367a^7 - 1738/367a^6 - 725/367a^5 + 1517/367a^4 + 228/367a^3 - 889/367a^2 - 181/367a + 32/367, 32/367a^13 - 93/367a^12 - 22/367a^11 + 448/367a^10 - 692/367a^9 + 183/367a^8 + 490/367a^7 - 876/367a^6 + 176/367a^5 + 632/367a^4 - 190/367a^3 - 666/367a^2 + 212/367a + 218/367, 143/367a^13 - 817/367a^12 + 1622/367a^11 - 567/367a^10 - 2496/367a^9 + 2756/367a^8 + 974/367a^7 - 2309/367a^6 - 498/367a^5 + 1448/367a^4 + 229/367a^3 - 476/367a^2 - 16/367a - 58/367, 33/367a^13 - 245/367a^12 + 459/367a^11 + 95/367a^10 - 1310/367a^9 + 636/367a^8 + 1721/367a^7 - 1408/367a^6 - 1470/367a^5 + 1294/367a^4 + 1295/367a^3 - 618/367a^2 - 653/367a + 156/367	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:	$35.8148412318$	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 35.8148412318 \cdot 1}{2\cdot\sqrt{391066968923137}}\cr\approx \mathstrut & 0.222867462546 \end{aligned}$

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

x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 5*x^12 + 4*x^11 - 14*x^10 + 5*x^9 + 13*x^8 - 9*x^7 - 9*x^6 + 8*x^5 + 4*x^4 - 5*x^3 + x - 1)

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 - 4*x^13 + 5*x^12 + 4*x^11 - 14*x^10 + 5*x^9 + 13*x^8 - 9*x^7 - 9*x^6 + 8*x^5 + 4*x^4 - 5*x^3 + x - 1, 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 - 4*x^13 + 5*x^12 + 4*x^11 - 14*x^10 + 5*x^9 + 13*x^8 - 9*x^7 - 9*x^6 + 8*x^5 + 4*x^4 - 5*x^3 + x - 1);

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 - 4*x^13 + 5*x^12 + 4*x^11 - 14*x^10 + 5*x^9 + 13*x^8 - 9*x^7 - 9*x^6 + 8*x^5 + 4*x^4 - 5*x^3 + x - 1);

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

$C_2^7.\GL(3,2)$ (as 14T51):

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 21504

The 48 conjugacy class representatives for

C_2^7.\GL(3,2)

Character table for

C_2^7.\GL(3,2)

Intermediate fields

7.3.2007889.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

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 14 sibling:	data not computed
Degree 28 siblings:	data not computed
Degree 42 siblings:	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	${\href{/padicField/2.7.0.1}{7} }^{2}$	${\href{/padicField/3.7.0.1}{7} }^{2}$	${\href{/padicField/5.14.0.1}{14} }$	${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$	${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$	R	${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$	${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$	${\href{/padicField/23.14.0.1}{14} }$	${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }$	${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }$	${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }$	${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$	${\href{/padicField/43.7.0.1}{7} }^{2}$	${\href{/padicField/47.7.0.1}{7} }^{2}$	${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$	${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$

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
$13$ 13	$\Q_{13}$	$x + 11$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{13}$	$x + 11$	$1$	$1$	$0$	Trivial	$[\ ]$
	13.2.2.2a1.1	$x^{4} + 24 x^{3} + 148 x^{2} + 61 x + 4$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
	13.2.2.2a1.1	$x^{4} + 24 x^{3} + 148 x^{2} + 61 x + 4$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
	13.4.1.0a1.1	$x^{4} + 3 x^{2} + 12 x + 2$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
$97$ 97	97.1.2.1a1.1	$x^{2} + 97$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	97.6.1.0a1.1	$x^{6} + 92 x^{3} + 58 x^{2} + 88 x + 5$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	97.6.1.0a1.1	$x^{6} + 92 x^{3} + 58 x^{2} + 88 x + 5$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
$109$ 109	$\Q_{109}$	$x + 103$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{109}$	$x + 103$	$1$	$1$	$0$	Trivial	$[\ ]$
	109.2.1.0a1.1	$x^{2} + 108 x + 6$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	109.2.1.0a1.1	$x^{2} + 108 x + 6$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	109.4.2.4a1.2	$x^{8} + 22 x^{6} + 196 x^{5} + 133 x^{4} + 2156 x^{3} + 9736 x^{2} + 1176 x + 145$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$

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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	14.2.391066968923137.1

Degree	$14$
Signature	$[2, 6]$
Discriminant	$3.911\times 10^{14}$
Root discriminant	$11.02$
Ramified primes	$13,97,109$
Class number	$1$
Class group	trivial
Galois group	$C_2^7.\GL(3,2)$ (as 14T51)

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 $a$ )