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Properties

Label 21.1.23354215149...5728.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{26}\cdot 3^{19}\cdot 7^{17}\cdot 23^{19}\cdot 29^{7}$
Root discriminant $1614.42$
Ramified primes $2, 3, 7, 23, 29$
Class number Not computed
Class group Not computed
Galois group $S_3\times F_7$ (as 21T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-715092588417024, 0, 0, 0, 0, 0, 0, 164940880734306, 0, 0, 0, 0, 0, 0, -21173202, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 21173202*x^14 + 164940880734306*x^7 - 715092588417024)
 
gp: K = bnfinit(x^21 - 21173202*x^14 + 164940880734306*x^7 - 715092588417024, 1)
 

Normalized defining polynomial

\( x^{21} - 21173202 x^{14} + 164940880734306 x^{7} - 715092588417024 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(23354215149520145770884611145212717381599630651700115392847547465728=2^{26}\cdot 3^{19}\cdot 7^{17}\cdot 23^{19}\cdot 29^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1614.42$
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:  $2, 3, 7, 23, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5}$, $\frac{1}{21} a^{6} - \frac{2}{21} a^{5} - \frac{1}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{483} a^{7} - \frac{2}{7}$, $\frac{1}{483} a^{8} - \frac{2}{7} a$, $\frac{1}{483} a^{9} - \frac{2}{7} a^{2}$, $\frac{1}{1449} a^{10} - \frac{3}{7} a^{3}$, $\frac{1}{10143} a^{11} + \frac{1}{10143} a^{10} + \frac{1}{3381} a^{9} - \frac{1}{3381} a^{8} + \frac{1}{1127} a^{7} - \frac{24}{49} a^{4} - \frac{24}{49} a^{3} - \frac{23}{49} a^{2} + \frac{23}{49} a - \frac{20}{49}$, $\frac{1}{10143} a^{12} + \frac{2}{10143} a^{10} - \frac{2}{3381} a^{9} - \frac{1}{1127} a^{8} - \frac{1}{1127} a^{7} - \frac{23}{147} a^{5} + \frac{1}{49} a^{3} - \frac{3}{49} a^{2} + \frac{20}{49} a + \frac{20}{49}$, $\frac{1}{10143} a^{13} - \frac{1}{10143} a^{10} + \frac{2}{3381} a^{9} - \frac{1}{3381} a^{8} + \frac{1}{3381} a^{7} - \frac{2}{147} a^{6} + \frac{1}{21} a^{5} - \frac{3}{7} a^{4} + \frac{17}{49} a^{3} + \frac{17}{49} a^{2} - \frac{5}{49} a - \frac{16}{49}$, $\frac{1}{7929431553897882} a^{14} - \frac{3447700390}{6384405437921} a^{7} - \frac{7685980390}{277582845127}$, $\frac{1}{47576589323387292} a^{15} + \frac{29311590991}{114919297882578} a^{8} - \frac{364578209839}{1665497070762} a$, $\frac{1}{11989300509493597584} a^{16} - \frac{1}{111012041754570348} a^{15} + \frac{1}{18502006959095058} a^{14} - \frac{24953144470439}{28959663066409656} a^{9} - \frac{29311590991}{268145028392682} a^{8} + \frac{29311590991}{44690838065447} a^{7} - \frac{158586799932229}{419705261832024} a^{2} - \frac{1300918860923}{3886159831778} a + \frac{16596750991}{1943079915889}$, $\frac{1}{215807409170884756512} a^{17} - \frac{1}{111012041754570348} a^{15} - \frac{1}{18502006959095058} a^{14} + \frac{94962644624425}{521273935195373808} a^{10} - \frac{29311590991}{268145028392682} a^{8} - \frac{29311590991}{44690838065447} a^{7} + \frac{860697407374115}{7554694712976432} a^{3} - \frac{1300918860923}{3886159831778} a - \frac{16596750991}{1943079915889}$, $\frac{1}{59562844931164192797312} a^{18} - \frac{1}{166518062631855522} a^{15} + \frac{1}{18502006959095058} a^{14} - \frac{6637452371701511}{143871606113923171008} a^{11} + \frac{1}{3381} a^{10} + \frac{1}{1127} a^{9} - \frac{386203820440}{402217542589023} a^{8} - \frac{110338687832}{134072514196341} a^{7} + \frac{85195758795090611}{2085095740781495232} a^{4} - \frac{23}{49} a^{3} - \frac{20}{49} a^{2} + \frac{1078362668737}{5829239747667} a + \frac{690726517728}{1943079915889}$, $\frac{1}{1072131208760955470351616} a^{19} + \frac{1}{111012041754570348} a^{15} - \frac{1}{27753010438642587} a^{14} + \frac{35915521929961657}{2589688910050617078144} a^{12} + \frac{1}{3381} a^{10} - \frac{1}{1127} a^{9} - \frac{49997793331}{268145028392682} a^{8} - \frac{32759291381}{44690838065447} a^{7} + \frac{3276668831419828211}{37531723334066914176} a^{5} - \frac{23}{49} a^{3} + \frac{20}{49} a^{2} - \frac{761125131449}{3886159831778} a + \frac{253300113746}{1943079915889}$, $\frac{1}{38596723515394396932658176} a^{20} - \frac{1}{111012041754570348} a^{15} - \frac{1}{27753010438642587} a^{14} - \frac{3283216473599765447}{93228800761822214813184} a^{13} + \frac{1}{10143} a^{10} - \frac{2}{3381} a^{9} + \frac{49997793331}{268145028392682} a^{8} - \frac{32759291381}{44690838065447} a^{7} + \frac{11702157743149135475}{1351142040026408910336} a^{6} - \frac{1}{21} a^{5} + \frac{3}{7} a^{4} - \frac{17}{49} a^{3} - \frac{17}{49} a^{2} - \frac{904371939313}{3886159831778} a - \frac{24282731381}{1943079915889}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_7$ (as 21T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 252
The 21 conjugacy class representatives for $S_3\times F_7$
Character table for $S_3\times F_7$ is not computed

Intermediate fields

3.1.392196.1, 7.1.116081956281751488.2

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

Sibling fields

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.14.20.8$x^{14} + 2 x^{13} + 2 x^{11} + 2 x^{8} + 2 x^{7} + 2 x^{6} + 2$$14$$1$$20$$(C_7:C_3) \times C_2$$[2]_{7}^{3}$
$3$3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.14.13.2$x^{14} + 3$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
7Data not computed
$23$23.7.6.1$x^{7} - 23$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
23.14.13.2$x^{14} + 46$$14$$1$$13$$(C_7:C_3) \times C_2$$[\ ]_{14}^{3}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$