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Properties

Label 21.21.9414288180...6237.1
Degree $21$
Signature $[21, 0]$
Discriminant $7^{14}\cdot 173^{9}$
Root discriminant $33.31$
Ramified primes $7, 173$
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![1, 2, -44, -97, 571, 1442, -2254, -6710, 3175, 12809, -2021, -12509, 676, 6826, -158, -2119, 32, 362, -3, -31, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 31*x^19 - 3*x^18 + 362*x^17 + 32*x^16 - 2119*x^15 - 158*x^14 + 6826*x^13 + 676*x^12 - 12509*x^11 - 2021*x^10 + 12809*x^9 + 3175*x^8 - 6710*x^7 - 2254*x^6 + 1442*x^5 + 571*x^4 - 97*x^3 - 44*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^21 - 31*x^19 - 3*x^18 + 362*x^17 + 32*x^16 - 2119*x^15 - 158*x^14 + 6826*x^13 + 676*x^12 - 12509*x^11 - 2021*x^10 + 12809*x^9 + 3175*x^8 - 6710*x^7 - 2254*x^6 + 1442*x^5 + 571*x^4 - 97*x^3 - 44*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{21} - 31 x^{19} - 3 x^{18} + 362 x^{17} + 32 x^{16} - 2119 x^{15} - 158 x^{14} + 6826 x^{13} + 676 x^{12} - 12509 x^{11} - 2021 x^{10} + 12809 x^{9} + 3175 x^{8} - 6710 x^{7} - 2254 x^{6} + 1442 x^{5} + 571 x^{4} - 97 x^{3} - 44 x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(94142881806955162927406195366237=7^{14}\cdot 173^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.31$
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:  $7, 173$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{14} a^{18} + \frac{3}{14} a^{17} + \frac{3}{14} a^{16} - \frac{1}{7} a^{15} - \frac{1}{14} a^{13} - \frac{3}{14} a^{12} + \frac{2}{7} a^{11} + \frac{3}{14} a^{9} + \frac{3}{7} a^{8} - \frac{3}{14} a^{7} + \frac{3}{14} a^{6} + \frac{5}{14} a^{4} - \frac{3}{14} a^{3} + \frac{2}{7} a^{2} - \frac{5}{14} a + \frac{3}{7}$, $\frac{1}{14} a^{19} + \frac{1}{14} a^{17} + \frac{3}{14} a^{16} - \frac{1}{14} a^{15} - \frac{1}{14} a^{14} + \frac{3}{7} a^{12} + \frac{1}{7} a^{11} + \frac{3}{14} a^{10} + \frac{2}{7} a^{9} - \frac{1}{2} a^{8} + \frac{5}{14} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} + \frac{3}{14} a^{4} - \frac{1}{14} a^{3} - \frac{3}{14} a^{2} + \frac{3}{14}$, $\frac{1}{162934304791286} a^{20} - \frac{88026754681}{162934304791286} a^{19} + \frac{770132756654}{81467152395643} a^{18} - \frac{2607635156019}{11638164627949} a^{17} + \frac{1751212016901}{23276329255898} a^{16} - \frac{16170348231445}{162934304791286} a^{15} - \frac{31896696514475}{162934304791286} a^{14} + \frac{23609362288305}{162934304791286} a^{13} + \frac{27042954172520}{81467152395643} a^{12} + \frac{78251361463763}{162934304791286} a^{11} + \frac{51359584672889}{162934304791286} a^{10} - \frac{34741355338729}{162934304791286} a^{9} - \frac{37116913859171}{81467152395643} a^{8} + \frac{25558031868333}{162934304791286} a^{7} + \frac{25550390568197}{81467152395643} a^{6} + \frac{6305989792553}{81467152395643} a^{5} + \frac{14719118160162}{81467152395643} a^{4} + \frac{1046377285263}{23276329255898} a^{3} - \frac{31921751560399}{162934304791286} a^{2} - \frac{50861426584869}{162934304791286} a + \frac{20693501531267}{81467152395643}$

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:  \( 1629233271.9 \) (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

\(\Q(\zeta_{7})^+\), 7.7.12431698517.1

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: 7.7.12431698517.1
Degree 14 sibling: 14.14.26736653147041339876997.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$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}$
$173$173.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
173.6.3.1$x^{6} - 346 x^{4} + 29929 x^{2} - 129442925$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
173.6.3.1$x^{6} - 346 x^{4} + 29929 x^{2} - 129442925$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
173.6.3.1$x^{6} - 346 x^{4} + 29929 x^{2} - 129442925$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$