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Properties

Label 21.21.5849237178...8921.1
Degree $21$
Signature $[21, 0]$
Discriminant $3^{28}\cdot 29^{8}\cdot 59^{10}$
Root discriminant $108.77$
Ramified primes $3, 29, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_7$ (as 21T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-477, 18885, 65247, -1211172, -482093, 4026302, 1195047, -4787032, -1213360, 2868715, 630480, -979409, -184556, 199461, 31328, -24355, -3019, 1727, 151, -65, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 65*x^19 + 151*x^18 + 1727*x^17 - 3019*x^16 - 24355*x^15 + 31328*x^14 + 199461*x^13 - 184556*x^12 - 979409*x^11 + 630480*x^10 + 2868715*x^9 - 1213360*x^8 - 4787032*x^7 + 1195047*x^6 + 4026302*x^5 - 482093*x^4 - 1211172*x^3 + 65247*x^2 + 18885*x - 477)
 
gp: K = bnfinit(x^21 - 3*x^20 - 65*x^19 + 151*x^18 + 1727*x^17 - 3019*x^16 - 24355*x^15 + 31328*x^14 + 199461*x^13 - 184556*x^12 - 979409*x^11 + 630480*x^10 + 2868715*x^9 - 1213360*x^8 - 4787032*x^7 + 1195047*x^6 + 4026302*x^5 - 482093*x^4 - 1211172*x^3 + 65247*x^2 + 18885*x - 477, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} - 65 x^{19} + 151 x^{18} + 1727 x^{17} - 3019 x^{16} - 24355 x^{15} + 31328 x^{14} + 199461 x^{13} - 184556 x^{12} - 979409 x^{11} + 630480 x^{10} + 2868715 x^{9} - 1213360 x^{8} - 4787032 x^{7} + 1195047 x^{6} + 4026302 x^{5} - 482093 x^{4} - 1211172 x^{3} + 65247 x^{2} + 18885 x - 477 \)

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:  \(5849237178513966970769296632946293201918921=3^{28}\cdot 29^{8}\cdot 59^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $108.77$
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:  $3, 29, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{29} a^{14} - \frac{1}{29} a^{13} + \frac{4}{29} a^{12} + \frac{3}{29} a^{11} + \frac{4}{29} a^{10} - \frac{4}{29} a^{9} - \frac{5}{29} a^{8} + \frac{6}{29} a^{7} - \frac{5}{29} a^{6} + \frac{7}{29} a^{5} - \frac{6}{29} a^{4} + \frac{3}{29} a^{3} + \frac{7}{29} a^{2} - \frac{7}{29} a - \frac{13}{29}$, $\frac{1}{29} a^{15} + \frac{3}{29} a^{13} + \frac{7}{29} a^{12} + \frac{7}{29} a^{11} - \frac{9}{29} a^{9} + \frac{1}{29} a^{8} + \frac{1}{29} a^{7} + \frac{2}{29} a^{6} + \frac{1}{29} a^{5} - \frac{3}{29} a^{4} + \frac{10}{29} a^{3} + \frac{9}{29} a - \frac{13}{29}$, $\frac{1}{29} a^{16} + \frac{10}{29} a^{13} - \frac{5}{29} a^{12} - \frac{9}{29} a^{11} + \frac{8}{29} a^{10} + \frac{13}{29} a^{9} - \frac{13}{29} a^{8} + \frac{13}{29} a^{7} - \frac{13}{29} a^{6} + \frac{5}{29} a^{5} - \frac{1}{29} a^{4} - \frac{9}{29} a^{3} - \frac{12}{29} a^{2} + \frac{8}{29} a + \frac{10}{29}$, $\frac{1}{87} a^{17} + \frac{1}{87} a^{15} - \frac{1}{87} a^{14} + \frac{38}{87} a^{13} + \frac{4}{29} a^{12} + \frac{40}{87} a^{11} - \frac{2}{87} a^{10} - \frac{12}{29} a^{9} + \frac{11}{87} a^{8} + \frac{38}{87} a^{7} + \frac{11}{29} a^{6} + \frac{13}{29} a^{5} + \frac{25}{87} a^{4} + \frac{23}{87} a^{3} + \frac{6}{29} a^{2} + \frac{3}{29} a - \frac{5}{29}$, $\frac{1}{87} a^{18} + \frac{1}{87} a^{16} - \frac{1}{87} a^{15} - \frac{1}{87} a^{14} - \frac{12}{29} a^{13} - \frac{1}{3} a^{12} - \frac{32}{87} a^{11} - \frac{6}{29} a^{10} - \frac{7}{87} a^{9} - \frac{28}{87} a^{8} - \frac{9}{29} a^{7} - \frac{9}{29} a^{6} + \frac{13}{87} a^{5} - \frac{4}{87} a^{4} - \frac{4}{29} a^{3} - \frac{1}{29} a^{2} - \frac{1}{29} a - \frac{5}{29}$, $\frac{1}{2523} a^{19} + \frac{1}{841} a^{18} - \frac{3}{841} a^{17} + \frac{8}{2523} a^{16} - \frac{17}{2523} a^{15} + \frac{25}{2523} a^{14} - \frac{955}{2523} a^{13} - \frac{596}{2523} a^{12} - \frac{775}{2523} a^{11} - \frac{1169}{2523} a^{10} - \frac{1018}{2523} a^{9} + \frac{1168}{2523} a^{8} + \frac{955}{2523} a^{7} + \frac{553}{2523} a^{6} - \frac{385}{2523} a^{5} - \frac{1030}{2523} a^{4} + \frac{592}{2523} a^{3} + \frac{125}{841} a^{2} + \frac{46}{841} a + \frac{298}{841}$, $\frac{1}{48733085937169954333622637615669687999} a^{20} + \frac{5961886198935486654003309419047126}{48733085937169954333622637615669687999} a^{19} - \frac{270679401253402843721061305481899113}{48733085937169954333622637615669687999} a^{18} - \frac{82456352169465351764574142454198416}{16244361979056651444540879205223229333} a^{17} - \frac{55723744480027158430615732485952783}{48733085937169954333622637615669687999} a^{16} + \frac{519441370032663057820808231935129000}{48733085937169954333622637615669687999} a^{15} - \frac{184677073692664813303141563935140993}{16244361979056651444540879205223229333} a^{14} + \frac{18381963403704383523146164962134855480}{48733085937169954333622637615669687999} a^{13} + \frac{18548824888753546070998389360322631540}{48733085937169954333622637615669687999} a^{12} - \frac{2403502906050845757252716391167368247}{16244361979056651444540879205223229333} a^{11} - \frac{2221977153029635030859800683684835535}{48733085937169954333622637615669687999} a^{10} + \frac{22745964736344724703978054777008541299}{48733085937169954333622637615669687999} a^{9} + \frac{23846656200250512089184880553245042208}{48733085937169954333622637615669687999} a^{8} - \frac{1808220939692196988175442459263107811}{48733085937169954333622637615669687999} a^{7} - \frac{7654504845332016981988888974450284764}{16244361979056651444540879205223229333} a^{6} + \frac{2114488877528026465458544083156081870}{5414787326352217148180293068407743111} a^{5} + \frac{21058291723094244419599216078994822036}{48733085937169954333622637615669687999} a^{4} + \frac{3831344826844933256991401789988369893}{16244361979056651444540879205223229333} a^{3} + \frac{840838710517025589311881646209503188}{5414787326352217148180293068407743111} a^{2} - \frac{43192742411094287597455670228007935}{560150413070919015328995834662869977} a + \frac{76457866285203149459488883821361133}{5414787326352217148180293068407743111}$

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:  \( 3149865113890000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_7$ (as 21T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2520
The 9 conjugacy class representatives for $A_7$
Character table for $A_7$

Intermediate fields

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

Sibling fields

Degree 7 sibling: data not computed
Degree 15 siblings: 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{3}$ R ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{5}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ R

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.3.5.3$x^{3} + 12$$3$$1$$5$$S_3$$[5/2]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.6.10.1$x^{6} - 18$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
3.6.11.2$x^{6} + 15$$6$$1$$11$$D_{6}$$[5/2]_{2}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
59Data not computed