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Properties

Label 21.9.51760737730...3264.2
Degree $21$
Signature $[9, 6]$
Discriminant $2^{18}\cdot 3^{30}\cdot 17^{2}\cdot 57605311^{2}$
Root discriminant $62.50$
Ramified primes $2, 3, 17, 57605311$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T143

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-216, -1080, -1800, -784, 576, 72, -776, -588, -114, 82, -144, -204, -21, 27, 15, -58, -36, 15, 10, 6, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 + 6*x^19 + 10*x^18 + 15*x^17 - 36*x^16 - 58*x^15 + 15*x^14 + 27*x^13 - 21*x^12 - 204*x^11 - 144*x^10 + 82*x^9 - 114*x^8 - 588*x^7 - 776*x^6 + 72*x^5 + 576*x^4 - 784*x^3 - 1800*x^2 - 1080*x - 216)
 
gp: K = bnfinit(x^21 - 6*x^20 + 6*x^19 + 10*x^18 + 15*x^17 - 36*x^16 - 58*x^15 + 15*x^14 + 27*x^13 - 21*x^12 - 204*x^11 - 144*x^10 + 82*x^9 - 114*x^8 - 588*x^7 - 776*x^6 + 72*x^5 + 576*x^4 - 784*x^3 - 1800*x^2 - 1080*x - 216, 1)
 

Normalized defining polynomial

\( x^{21} - 6 x^{20} + 6 x^{19} + 10 x^{18} + 15 x^{17} - 36 x^{16} - 58 x^{15} + 15 x^{14} + 27 x^{13} - 21 x^{12} - 204 x^{11} - 144 x^{10} + 82 x^{9} - 114 x^{8} - 588 x^{7} - 776 x^{6} + 72 x^{5} + 576 x^{4} - 784 x^{3} - 1800 x^{2} - 1080 x - 216 \)

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:  \(51760737730687241530618054097890443264=2^{18}\cdot 3^{30}\cdot 17^{2}\cdot 57605311^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.50$
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:  $2, 3, 17, 57605311$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{13} - \frac{1}{2} a^{12} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{68} a^{18} + \frac{3}{34} a^{17} - \frac{4}{17} a^{16} + \frac{4}{17} a^{15} - \frac{7}{68} a^{14} - \frac{3}{17} a^{13} + \frac{25}{68} a^{11} - \frac{11}{68} a^{10} + \frac{13}{68} a^{9} - \frac{6}{17} a^{8} - \frac{3}{34} a^{7} - \frac{1}{2} a^{6} + \frac{3}{17} a^{5} + \frac{4}{17} a^{4} - \frac{3}{17} a^{3} + \frac{5}{17} a^{2} - \frac{2}{17} a - \frac{1}{17}$, $\frac{1}{612} a^{19} + \frac{11}{204} a^{17} + \frac{5}{306} a^{16} - \frac{23}{204} a^{15} - \frac{2}{17} a^{14} - \frac{13}{612} a^{13} + \frac{33}{68} a^{12} + \frac{1}{68} a^{11} - \frac{35}{102} a^{10} - \frac{1}{12} a^{9} + \frac{21}{68} a^{8} - \frac{67}{306} a^{7} + \frac{6}{17} a^{6} + \frac{19}{102} a^{5} - \frac{37}{306} a^{4} + \frac{2}{51} a^{3} + \frac{4}{17} a^{2} + \frac{11}{153} a + \frac{19}{51}$, $\frac{1}{5508} a^{20} - \frac{1}{1836} a^{19} - \frac{1}{1836} a^{18} + \frac{1}{5508} a^{17} - \frac{151}{612} a^{16} - \frac{151}{612} a^{15} + \frac{1373}{5508} a^{14} - \frac{229}{918} a^{13} - \frac{151}{306} a^{12} - \frac{889}{1836} a^{11} - \frac{899}{1836} a^{10} + \frac{13}{51} a^{9} + \frac{163}{5508} a^{8} - \frac{167}{918} a^{7} + \frac{319}{918} a^{6} - \frac{271}{2754} a^{5} - \frac{259}{918} a^{4} - \frac{37}{153} a^{3} + \frac{182}{1377} a^{2} + \frac{32}{459} a + \frac{2}{153}$

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:  \( 223805105406 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T143:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14224896
The 88 conjugacy class representatives for t21n143 are not computed
Character table for t21n143 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\)

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

Sibling fields

Degree 24 sibling: 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 R R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $21$ $21$ R ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.18.36$x^{12} - 6 x^{11} + 6 x^{10} + 8 x^{9} + 8 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8$$4$$3$$18$12T99$[2, 2, 2, 2, 2]^{6}$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
3.12.18.74$x^{12} - 6 x^{11} - 3 x^{10} - 12 x^{9} + 9 x^{8} + 9 x^{7} + 12 x^{6} - 9 x^{3} - 9$$6$$2$$18$$C_{12}$$[2]_{2}^{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.7.0.1$x^{7} - x + 3$$1$$7$$0$$C_7$$[\ ]^{7}$
57605311Data not computed