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Properties

Label 21.5.15976350183...1441.1
Degree $21$
Signature $[5, 8]$
Discriminant $11^{14}\cdot 29^{10}$
Root discriminant $24.58$
Ramified primes $11, 29$
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![1, -2, -2, 25, -16, -32, 116, -41, -76, 228, -106, -70, 277, -124, -69, 153, -37, -25, 26, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 2*x^19 + 26*x^18 - 25*x^17 - 37*x^16 + 153*x^15 - 69*x^14 - 124*x^13 + 277*x^12 - 70*x^11 - 106*x^10 + 228*x^9 - 76*x^8 - 41*x^7 + 116*x^6 - 32*x^5 - 16*x^4 + 25*x^3 - 2*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^21 - 2*x^20 - 2*x^19 + 26*x^18 - 25*x^17 - 37*x^16 + 153*x^15 - 69*x^14 - 124*x^13 + 277*x^12 - 70*x^11 - 106*x^10 + 228*x^9 - 76*x^8 - 41*x^7 + 116*x^6 - 32*x^5 - 16*x^4 + 25*x^3 - 2*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{21} - 2 x^{20} - 2 x^{19} + 26 x^{18} - 25 x^{17} - 37 x^{16} + 153 x^{15} - 69 x^{14} - 124 x^{13} + 277 x^{12} - 70 x^{11} - 106 x^{10} + 228 x^{9} - 76 x^{8} - 41 x^{7} + 116 x^{6} - 32 x^{5} - 16 x^{4} + 25 x^{3} - 2 x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(159763501833017076073675531441=11^{14}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.58$
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:  $11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} - \frac{4}{11} a^{12} + \frac{2}{11} a^{11} - \frac{3}{11} a^{10} - \frac{4}{11} a^{9} + \frac{1}{11} a^{8} - \frac{2}{11} a^{7} + \frac{5}{11} a^{6} - \frac{1}{11} a^{5} + \frac{1}{11} a^{4} + \frac{2}{11} a^{3} + \frac{5}{11} a^{2} - \frac{4}{11} a - \frac{4}{11}$, $\frac{1}{11} a^{14} - \frac{3}{11} a^{12} + \frac{5}{11} a^{11} - \frac{5}{11} a^{10} - \frac{4}{11} a^{9} + \frac{2}{11} a^{8} - \frac{3}{11} a^{7} - \frac{3}{11} a^{6} - \frac{3}{11} a^{5} - \frac{5}{11} a^{4} + \frac{2}{11} a^{3} + \frac{5}{11} a^{2} + \frac{2}{11} a - \frac{5}{11}$, $\frac{1}{11} a^{15} + \frac{4}{11} a^{12} + \frac{1}{11} a^{11} - \frac{2}{11} a^{10} + \frac{1}{11} a^{9} + \frac{2}{11} a^{7} + \frac{1}{11} a^{6} + \frac{3}{11} a^{5} + \frac{5}{11} a^{4} - \frac{5}{11} a^{2} + \frac{5}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{16} - \frac{5}{11} a^{12} + \frac{1}{11} a^{11} + \frac{2}{11} a^{10} + \frac{5}{11} a^{9} - \frac{2}{11} a^{8} - \frac{2}{11} a^{7} + \frac{5}{11} a^{6} - \frac{2}{11} a^{5} - \frac{4}{11} a^{4} - \frac{2}{11} a^{3} - \frac{4}{11} a^{2} + \frac{4}{11} a + \frac{5}{11}$, $\frac{1}{11} a^{17} + \frac{3}{11} a^{12} + \frac{1}{11} a^{11} + \frac{1}{11} a^{10} + \frac{3}{11} a^{8} - \frac{5}{11} a^{7} + \frac{1}{11} a^{6} + \frac{2}{11} a^{5} + \frac{3}{11} a^{4} - \frac{5}{11} a^{3} - \frac{4}{11} a^{2} - \frac{4}{11} a + \frac{2}{11}$, $\frac{1}{11} a^{18} + \frac{2}{11} a^{12} - \frac{5}{11} a^{11} - \frac{2}{11} a^{10} + \frac{4}{11} a^{9} + \frac{3}{11} a^{8} - \frac{4}{11} a^{7} - \frac{2}{11} a^{6} - \frac{5}{11} a^{5} + \frac{3}{11} a^{4} + \frac{1}{11} a^{3} + \frac{3}{11} a^{2} + \frac{3}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{19} + \frac{3}{11} a^{12} + \frac{5}{11} a^{11} - \frac{1}{11} a^{10} + \frac{5}{11} a^{8} + \frac{2}{11} a^{7} - \frac{4}{11} a^{6} + \frac{5}{11} a^{5} - \frac{1}{11} a^{4} - \frac{1}{11} a^{3} + \frac{4}{11} a^{2} - \frac{2}{11} a - \frac{3}{11}$, $\frac{1}{34381523} a^{20} - \frac{347684}{34381523} a^{19} + \frac{1159211}{34381523} a^{18} - \frac{958305}{34381523} a^{17} + \frac{310778}{34381523} a^{16} - \frac{166623}{34381523} a^{15} - \frac{1048216}{34381523} a^{14} - \frac{1434150}{34381523} a^{13} - \frac{16464672}{34381523} a^{12} - \frac{3479431}{34381523} a^{11} - \frac{873983}{3125593} a^{10} + \frac{9321230}{34381523} a^{9} + \frac{9725278}{34381523} a^{8} - \frac{3216749}{34381523} a^{7} + \frac{2912924}{34381523} a^{6} - \frac{13472599}{34381523} a^{5} - \frac{5411929}{34381523} a^{4} + \frac{1433411}{34381523} a^{3} - \frac{1650613}{34381523} a^{2} + \frac{1423927}{34381523} a - \frac{4860760}{34381523}$

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:  $12$
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:  \( 2549812.08882 \) (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: 7.3.12313081.1
Degree 15 siblings: Deg 15, Deg 15
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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.3.2.1$x^{3} - 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.12.10.3$x^{12} + 220 x^{6} + 41503$$6$$2$$10$$C_3 : C_4$$[\ ]_{6}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.12.8.1$x^{12} - 87 x^{9} + 2523 x^{6} - 24389 x^{3} + 4851240379$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$