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Properties

Label 21.9.11159702690...6281.1
Degree $21$
Signature $[9, 6]$
Discriminant $13^{16}\cdot 109^{6}$
Root discriminant $26.97$
Ramified primes $13, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times \PSL(2,7)$ (as 21T22)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-53, 93, 193, 195, -669, -1087, -570, 613, 2635, 1430, 1998, -466, 465, -127, 7, 288, -152, -24, 31, -10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 10*x^19 + 31*x^18 - 24*x^17 - 152*x^16 + 288*x^15 + 7*x^14 - 127*x^13 + 465*x^12 - 466*x^11 + 1998*x^10 + 1430*x^9 + 2635*x^8 + 613*x^7 - 570*x^6 - 1087*x^5 - 669*x^4 + 195*x^3 + 193*x^2 + 93*x - 53)
 
gp: K = bnfinit(x^21 - 2*x^20 - 10*x^19 + 31*x^18 - 24*x^17 - 152*x^16 + 288*x^15 + 7*x^14 - 127*x^13 + 465*x^12 - 466*x^11 + 1998*x^10 + 1430*x^9 + 2635*x^8 + 613*x^7 - 570*x^6 - 1087*x^5 - 669*x^4 + 195*x^3 + 193*x^2 + 93*x - 53, 1)
 

Normalized defining polynomial

\( x^{21} - 2 x^{20} - 10 x^{19} + 31 x^{18} - 24 x^{17} - 152 x^{16} + 288 x^{15} + 7 x^{14} - 127 x^{13} + 465 x^{12} - 466 x^{11} + 1998 x^{10} + 1430 x^{9} + 2635 x^{8} + 613 x^{7} - 570 x^{6} - 1087 x^{5} - 669 x^{4} + 195 x^{3} + 193 x^{2} + 93 x - 53 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1115970269016553289813936756281=13^{16}\cdot 109^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.97$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $13, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{91} a^{18} - \frac{3}{7} a^{17} + \frac{19}{91} a^{16} + \frac{25}{91} a^{15} - \frac{34}{91} a^{14} + \frac{20}{91} a^{13} - \frac{16}{91} a^{12} + \frac{36}{91} a^{11} - \frac{3}{13} a^{10} - \frac{2}{7} a^{9} + \frac{45}{91} a^{8} - \frac{17}{91} a^{7} + \frac{23}{91} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} + \frac{30}{91} a^{3} - \frac{3}{7} a^{2} + \frac{45}{91} a - \frac{34}{91}$, $\frac{1}{2821} a^{19} + \frac{12}{2821} a^{18} - \frac{787}{2821} a^{17} - \frac{1}{403} a^{16} + \frac{58}{2821} a^{15} - \frac{258}{2821} a^{14} + \frac{549}{2821} a^{13} + \frac{1}{7} a^{12} + \frac{814}{2821} a^{11} - \frac{1097}{2821} a^{10} - \frac{183}{403} a^{9} - \frac{907}{2821} a^{8} + \frac{1158}{2821} a^{7} - \frac{1245}{2821} a^{6} - \frac{83}{217} a^{5} + \frac{1356}{2821} a^{4} + \frac{148}{403} a^{3} + \frac{1059}{2821} a^{2} - \frac{145}{403} a - \frac{824}{2821}$, $\frac{1}{25006353731497176699208474054219} a^{20} + \frac{241112615084148571705637632}{3572336247356739528458353436317} a^{19} - \frac{102143917139276008402777142270}{25006353731497176699208474054219} a^{18} + \frac{7449328112654443117140104550668}{25006353731497176699208474054219} a^{17} - \frac{11404474528755787395557428268419}{25006353731497176699208474054219} a^{16} + \frac{1641532436888432041080912398212}{25006353731497176699208474054219} a^{15} + \frac{10484042046863781708727631375540}{25006353731497176699208474054219} a^{14} - \frac{3885425715994176304961429075043}{25006353731497176699208474054219} a^{13} + \frac{94726260148737950153321014213}{3572336247356739528458353436317} a^{12} - \frac{5626499710894136407172990839062}{25006353731497176699208474054219} a^{11} + \frac{7724012769405762688251529811028}{25006353731497176699208474054219} a^{10} - \frac{4857959118669862019201900002764}{25006353731497176699208474054219} a^{9} - \frac{7109793771972959040372975841024}{25006353731497176699208474054219} a^{8} - \frac{4194418937610566369617900301252}{25006353731497176699208474054219} a^{7} - \frac{4315965996778192541461172751482}{25006353731497176699208474054219} a^{6} + \frac{626128032786352072232834880696}{25006353731497176699208474054219} a^{5} + \frac{1420868512292147003680867728135}{3572336247356739528458353436317} a^{4} + \frac{10291304998882618078663981009910}{25006353731497176699208474054219} a^{3} - \frac{7782814839655357219059489017642}{25006353731497176699208474054219} a^{2} + \frac{7400382629978737062935451565822}{25006353731497176699208474054219} a - \frac{5843224372491922452979579657270}{25006353731497176699208474054219}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

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

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $14$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 12045126.3727 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times \PSL(2,7)$ (as 21T22):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 504
The 18 conjugacy class representatives for $C_3\times \PSL(2,7)$
Character table for $C_3\times \PSL(2,7)$

Intermediate fields

3.3.169.1, 7.3.2007889.2

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

Sibling fields

Degree 24 siblings: data not computed
Degree 42 siblings: data not computed
Arithmetically equvalently 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 $21$ $21$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.12.10.5$x^{12} + 65 x^{6} + 1352$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$