# Properties

 Label 21.3.647...536.1 Degree $21$ Signature $[3, 9]$ Discriminant $-6.478\times 10^{35}$ Root discriminant $50.73$ Ramified primes $2, 79, 159017$ Class number $1$ (GRH) Class group trivial (GRH) Galois group 21T62

# Learn more

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 + 22*x^19 - 44*x^18 + 46*x^17 - 18*x^16 - 9*x^15 - 94*x^14 + 67*x^13 + 50*x^12 + 44*x^11 - 96*x^10 + 288*x^9 + 690*x^8 + 639*x^7 - 246*x^6 - 1218*x^5 - 1692*x^4 - 1384*x^3 - 784*x^2 - 288*x - 64)

gp: K = bnfinit(x^21 - 6*x^20 + 22*x^19 - 44*x^18 + 46*x^17 - 18*x^16 - 9*x^15 - 94*x^14 + 67*x^13 + 50*x^12 + 44*x^11 - 96*x^10 + 288*x^9 + 690*x^8 + 639*x^7 - 246*x^6 - 1218*x^5 - 1692*x^4 - 1384*x^3 - 784*x^2 - 288*x - 64, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-64, -288, -784, -1384, -1692, -1218, -246, 639, 690, 288, -96, 44, 50, 67, -94, -9, -18, 46, -44, 22, -6, 1]);

$$x^{21} - 6 x^{20} + 22 x^{19} - 44 x^{18} + 46 x^{17} - 18 x^{16} - 9 x^{15} - 94 x^{14} + 67 x^{13} + 50 x^{12} + 44 x^{11} - 96 x^{10} + 288 x^{9} + 690 x^{8} + 639 x^{7} - 246 x^{6} - 1218 x^{5} - 1692 x^{4} - 1384 x^{3} - 784 x^{2} - 288 x - 64$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $21$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[3, 9]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-647754154086026198212292912701505536$$$$\medspace = -\,2^{23}\cdot 79^{7}\cdot 159017^{3}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $50.73$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $2, 79, 159017$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{3}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{3}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{3}{16} a^{8} + \frac{3}{16} a^{7} - \frac{5}{16} a^{6} - \frac{3}{16} a^{5} + \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{15} + \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{3}{16} a^{9} + \frac{1}{8} a^{6} + \frac{3}{16} a^{5} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{16} + \frac{1}{32} a^{12} + \frac{1}{16} a^{11} + \frac{1}{32} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{7}{16} a^{7} + \frac{11}{32} a^{6} + \frac{1}{8} a^{5} - \frac{3}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{17} + \frac{1}{32} a^{13} - \frac{1}{16} a^{12} + \frac{1}{32} a^{11} - \frac{1}{8} a^{10} - \frac{3}{16} a^{8} - \frac{13}{32} a^{7} - \frac{1}{4} a^{6} + \frac{1}{16} a^{5} + \frac{3}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{1984} a^{18} + \frac{1}{1984} a^{17} - \frac{15}{992} a^{16} - \frac{3}{124} a^{15} + \frac{9}{1984} a^{14} + \frac{11}{1984} a^{13} + \frac{85}{1984} a^{12} + \frac{5}{1984} a^{11} + \frac{117}{992} a^{10} - \frac{131}{992} a^{9} + \frac{485}{1984} a^{8} - \frac{949}{1984} a^{7} - \frac{91}{496} a^{6} + \frac{189}{992} a^{5} - \frac{137}{496} a^{4} + \frac{9}{62} a^{3} - \frac{1}{2} a^{2} - \frac{7}{62} a + \frac{11}{31}$, $\frac{1}{7936} a^{19} - \frac{1}{256} a^{17} + \frac{53}{3968} a^{16} + \frac{181}{7936} a^{15} - \frac{123}{3968} a^{14} - \frac{211}{3968} a^{13} - \frac{113}{1984} a^{12} - \frac{887}{7936} a^{11} - \frac{7}{64} a^{10} - \frac{617}{7936} a^{9} - \frac{841}{3968} a^{8} - \frac{2639}{7936} a^{7} + \frac{1921}{3968} a^{6} - \frac{525}{3968} a^{5} + \frac{581}{1984} a^{4} + \frac{119}{992} a^{3} - \frac{45}{496} a^{2} - \frac{33}{248} a + \frac{51}{124}$, $\frac{1}{31744} a^{20} - \frac{1}{31744} a^{19} + \frac{1}{31744} a^{18} - \frac{327}{31744} a^{17} - \frac{389}{31744} a^{16} + \frac{21}{31744} a^{15} - \frac{55}{1984} a^{14} + \frac{409}{15872} a^{13} + \frac{797}{31744} a^{12} - \frac{317}{31744} a^{11} - \frac{1685}{31744} a^{10} + \frac{471}{31744} a^{9} + \frac{4643}{31744} a^{8} - \frac{11487}{31744} a^{7} - \frac{539}{7936} a^{6} - \frac{5161}{15872} a^{5} + \frac{3953}{7936} a^{4} - \frac{1785}{3968} a^{3} - \frac{269}{1984} a^{2} - \frac{89}{992} a - \frac{195}{496}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

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

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $11$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$182519624586$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{3}\cdot(2\pi)^{9}\cdot 182519624586 \cdot 1}{2\sqrt{647754154086026198212292912701505536}}\approx 13.8446910894312$ (assuming GRH)

## Galois group

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 16464 The 65 conjugacy class representatives for t21n62 are not computed Character table for t21n62 is not computed

## Intermediate fields

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

## Sibling fields

 Degree 28 sibling: data not computed Degree 42 siblings: data not computed

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$ $21$ $21$ ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.7.0.1}{7} }$ ${\href{/padicField/13.6.0.1}{6} }^{3}{,}\,{\href{/padicField/13.3.0.1}{3} }$ ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.7.0.1}{7} }$ ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.7.0.1}{7} }$ ${\href{/padicField/31.2.0.1}{2} }^{10}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.7.0.1}{7} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }$ $21$

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

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

gp: idealfactors = idealprimedec(K, p); \\ get the data

gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

magma: idealfactors := Factorization(p*Integers(K)); // get the data

magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$$\Q_{2}$$x + 1$$1$$1$$0Trivial[\ ] 2.2.3.3x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3] 2.2.2.1x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3] 2.4.4.1x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2} 2.4.4.1x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$79$79.7.0.1$x^{7} - x + 9$$1$$7$$0$$C_7$$[\ ]^{7} 79.14.7.1x^{14} - 986078 x^{8} + 243087455521 x^{2} - 1555516627878879$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$159017$Data not computed