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Properties

Label 21.11.3913912048...3159.1
Degree $21$
Signature $[11, 5]$
Discriminant $-\,7^{5}\cdot 17^{8}\cdot 2017^{5}$
Root discriminant $28.63$
Ramified primes $7, 17, 2017$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T38

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -6, -53, -58, 243, 242, -345, 158, 534, -136, -294, -132, 170, 7, -79, 8, 25, 12, -9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 9*x^19 + 12*x^18 + 25*x^17 + 8*x^16 - 79*x^15 + 7*x^14 + 170*x^13 - 132*x^12 - 294*x^11 - 136*x^10 + 534*x^9 + 158*x^8 - 345*x^7 + 242*x^6 + 243*x^5 - 58*x^4 - 53*x^3 - 6*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^21 - 2*x^20 - 9*x^19 + 12*x^18 + 25*x^17 + 8*x^16 - 79*x^15 + 7*x^14 + 170*x^13 - 132*x^12 - 294*x^11 - 136*x^10 + 534*x^9 + 158*x^8 - 345*x^7 + 242*x^6 + 243*x^5 - 58*x^4 - 53*x^3 - 6*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{21} - 2 x^{20} - 9 x^{19} + 12 x^{18} + 25 x^{17} + 8 x^{16} - 79 x^{15} + 7 x^{14} + 170 x^{13} - 132 x^{12} - 294 x^{11} - 136 x^{10} + 534 x^{9} + 158 x^{8} - 345 x^{7} + 242 x^{6} + 243 x^{5} - 58 x^{4} - 53 x^{3} - 6 x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-3913912048328213377158221353159=-\,7^{5}\cdot 17^{8}\cdot 2017^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.63$
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:  $7, 17, 2017$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{51} a^{19} - \frac{1}{3} a^{18} - \frac{10}{51} a^{17} - \frac{25}{51} a^{16} + \frac{2}{51} a^{15} + \frac{1}{17} a^{14} - \frac{8}{17} a^{13} + \frac{7}{51} a^{12} - \frac{13}{51} a^{11} + \frac{5}{51} a^{10} + \frac{1}{51} a^{9} - \frac{1}{17} a^{8} + \frac{1}{3} a^{7} + \frac{8}{51} a^{6} - \frac{23}{51} a^{5} + \frac{6}{17} a^{4} - \frac{4}{51} a^{3} - \frac{16}{51} a^{2} - \frac{13}{51} a + \frac{1}{51}$, $\frac{1}{276266820308353926852243} a^{20} + \frac{1590189731017590564764}{276266820308353926852243} a^{19} - \frac{30500422167018097964723}{92088940102784642284081} a^{18} - \frac{102291791865290625998882}{276266820308353926852243} a^{17} - \frac{111894764051267492682467}{276266820308353926852243} a^{16} - \frac{23262080994744850410097}{276266820308353926852243} a^{15} - \frac{9760219397467728586546}{92088940102784642284081} a^{14} + \frac{68502529585440152237698}{276266820308353926852243} a^{13} + \frac{16103092142361477942839}{92088940102784642284081} a^{12} + \frac{121699421857170115322335}{276266820308353926852243} a^{11} - \frac{9734388459507645827529}{92088940102784642284081} a^{10} - \frac{89232117047012262642596}{276266820308353926852243} a^{9} - \frac{61850074257246569546695}{276266820308353926852243} a^{8} - \frac{2054593710730324551643}{12011600882971909863141} a^{7} + \frac{722157739387398945872}{4003866960990636621047} a^{6} + \frac{89449252064614374077080}{276266820308353926852243} a^{5} - \frac{31364481670765154148760}{276266820308353926852243} a^{4} + \frac{133538394105027662950588}{276266820308353926852243} a^{3} + \frac{115306907624482467283486}{276266820308353926852243} a^{2} + \frac{31434553065932889677413}{92088940102784642284081} a + \frac{89005797126827390633290}{276266820308353926852243}$

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:  $15$
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:  \( 47695486.7102 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T38:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5040
The 15 conjugacy class representatives for t21n38
Character table for t21n38

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 7 sibling: 7.5.4080391.1
Degree 14 sibling: Deg 14
Degree 30 sibling: data not computed
Degree 35 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ R ${\href{/LocalNumberField/19.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{3}$

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
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.5.0.1$x^{5} - x + 4$$1$$5$$0$$C_5$$[\ ]^{5}$
7.5.0.1$x^{5} - x + 4$$1$$5$$0$$C_5$$[\ ]^{5}$
7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
2017Data not computed