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Properties

Label 21.21.2406787169...5312.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{27}\cdot 7^{14}\cdot 31^{9}$
Root discriminant $38.87$
Ramified primes $2, 7, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_7$ (as 21T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -56, -276, 1052, 4492, -2788, -24311, -14531, 37753, 43350, -15763, -35382, -305, 14108, 1767, -3214, -473, 432, 51, -32, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 32*x^19 + 51*x^18 + 432*x^17 - 473*x^16 - 3214*x^15 + 1767*x^14 + 14108*x^13 - 305*x^12 - 35382*x^11 - 15763*x^10 + 43350*x^9 + 37753*x^8 - 14531*x^7 - 24311*x^6 - 2788*x^5 + 4492*x^4 + 1052*x^3 - 276*x^2 - 56*x + 8)
 
gp: K = bnfinit(x^21 - 2*x^20 - 32*x^19 + 51*x^18 + 432*x^17 - 473*x^16 - 3214*x^15 + 1767*x^14 + 14108*x^13 - 305*x^12 - 35382*x^11 - 15763*x^10 + 43350*x^9 + 37753*x^8 - 14531*x^7 - 24311*x^6 - 2788*x^5 + 4492*x^4 + 1052*x^3 - 276*x^2 - 56*x + 8, 1)
 

Normalized defining polynomial

\( x^{21} - 2 x^{20} - 32 x^{19} + 51 x^{18} + 432 x^{17} - 473 x^{16} - 3214 x^{15} + 1767 x^{14} + 14108 x^{13} - 305 x^{12} - 35382 x^{11} - 15763 x^{10} + 43350 x^{9} + 37753 x^{8} - 14531 x^{7} - 24311 x^{6} - 2788 x^{5} + 4492 x^{4} + 1052 x^{3} - 276 x^{2} - 56 x + 8 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2406787169604002863343075235725312=2^{27}\cdot 7^{14}\cdot 31^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.87$
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:  $2, 7, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{2} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4}$, $\frac{1}{1079578212093412} a^{20} - \frac{7949774923281}{1079578212093412} a^{19} + \frac{12715609328879}{269894553023353} a^{18} + \frac{184189865438001}{1079578212093412} a^{17} - \frac{84175519522045}{1079578212093412} a^{16} - \frac{21304611023181}{1079578212093412} a^{15} - \frac{227835955447891}{1079578212093412} a^{14} - \frac{5738938604513}{25106470048684} a^{13} + \frac{449140295544647}{1079578212093412} a^{12} - \frac{40692109083229}{1079578212093412} a^{11} + \frac{125643006466249}{1079578212093412} a^{10} + \frac{466747363165087}{1079578212093412} a^{9} + \frac{143955032155503}{1079578212093412} a^{8} - \frac{395339627893689}{1079578212093412} a^{7} + \frac{6779693814587}{539789106046706} a^{6} + \frac{85957087125313}{539789106046706} a^{5} + \frac{506684194022259}{1079578212093412} a^{4} - \frac{8811130171145}{539789106046706} a^{3} + \frac{120217707214026}{269894553023353} a^{2} - \frac{63940930981172}{269894553023353} a + \frac{80943118923147}{269894553023353}$

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:  $20$
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:  \( 12111978767.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_7$ (as 21T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.7.36622433792.1

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

Sibling fields

Galois closure: data not computed
Degree 7 sibling: 7.7.36622433792.1
Degree 14 sibling: Deg 14

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$31$31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.6.3.1$x^{6} - 62 x^{4} + 961 x^{2} - 2413071$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.1$x^{6} - 62 x^{4} + 961 x^{2} - 2413071$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.1$x^{6} - 62 x^{4} + 961 x^{2} - 2413071$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$