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Properties

Label 21.21.6569393369...6521.1
Degree $21$
Signature $[21, 0]$
Discriminant $7^{32}\cdot 29^{6}$
Root discriminant $50.77$
Ramified primes $7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T28

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 28, -154, -462, 6559, -18788, -1302, 109569, -212492, 63070, 331016, -570745, 424998, -134918, -16603, 26810, -6384, -630, 476, -42, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 42*x^19 + 476*x^18 - 630*x^17 - 6384*x^16 + 26810*x^15 - 16603*x^14 - 134918*x^13 + 424998*x^12 - 570745*x^11 + 331016*x^10 + 63070*x^9 - 212492*x^8 + 109569*x^7 - 1302*x^6 - 18788*x^5 + 6559*x^4 - 462*x^3 - 154*x^2 + 28*x - 1)
 
gp: K = bnfinit(x^21 - 7*x^20 - 42*x^19 + 476*x^18 - 630*x^17 - 6384*x^16 + 26810*x^15 - 16603*x^14 - 134918*x^13 + 424998*x^12 - 570745*x^11 + 331016*x^10 + 63070*x^9 - 212492*x^8 + 109569*x^7 - 1302*x^6 - 18788*x^5 + 6559*x^4 - 462*x^3 - 154*x^2 + 28*x - 1, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 42 x^{19} + 476 x^{18} - 630 x^{17} - 6384 x^{16} + 26810 x^{15} - 16603 x^{14} - 134918 x^{13} + 424998 x^{12} - 570745 x^{11} + 331016 x^{10} + 63070 x^{9} - 212492 x^{8} + 109569 x^{7} - 1302 x^{6} - 18788 x^{5} + 6559 x^{4} - 462 x^{3} - 154 x^{2} + 28 x - 1 \)

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:  \(656939336998075042895784450637466521=7^{32}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.77$
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:  $7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{3}{7} a^{9} + \frac{3}{7} a^{8} + \frac{2}{7} a^{7} - \frac{2}{7} a^{4} + \frac{2}{7} a^{3} + \frac{1}{7} a^{2} + \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{12} + \frac{2}{7} a^{10} - \frac{1}{7} a^{9} - \frac{2}{7} a^{8} + \frac{2}{7} a^{7} - \frac{2}{7} a^{5} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{10} - \frac{1}{7} a^{9} + \frac{3}{7} a^{8} + \frac{3}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{3} + \frac{2}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{14} + \frac{3}{7} a^{7} - \frac{3}{7}$, $\frac{1}{7} a^{15} + \frac{3}{7} a^{8} - \frac{3}{7} a$, $\frac{1}{7} a^{16} + \frac{3}{7} a^{9} - \frac{3}{7} a^{2}$, $\frac{1}{7} a^{17} + \frac{3}{7} a^{10} - \frac{3}{7} a^{3}$, $\frac{1}{7} a^{18} + \frac{3}{7} a^{10} - \frac{2}{7} a^{9} - \frac{2}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{497} a^{19} + \frac{34}{497} a^{18} + \frac{18}{497} a^{17} + \frac{20}{497} a^{16} + \frac{34}{497} a^{15} - \frac{4}{497} a^{14} + \frac{34}{497} a^{13} - \frac{4}{497} a^{12} - \frac{27}{497} a^{11} + \frac{35}{71} a^{10} + \frac{150}{497} a^{9} + \frac{41}{497} a^{8} + \frac{17}{71} a^{7} - \frac{33}{497} a^{6} + \frac{235}{497} a^{5} + \frac{170}{497} a^{4} - \frac{128}{497} a^{3} - \frac{44}{497} a^{2} + \frac{216}{497} a - \frac{194}{497}$, $\frac{1}{118446184548895651} a^{20} - \frac{39780275299035}{118446184548895651} a^{19} - \frac{1089462088114185}{118446184548895651} a^{18} - \frac{759140281659462}{16920883506985093} a^{17} - \frac{2258529781680261}{118446184548895651} a^{16} + \frac{2541515807364591}{118446184548895651} a^{15} - \frac{4413016376558438}{118446184548895651} a^{14} + \frac{1064090422715128}{16920883506985093} a^{13} - \frac{5414815471371506}{118446184548895651} a^{12} + \frac{4422944613686268}{118446184548895651} a^{11} + \frac{517243234115833}{118446184548895651} a^{10} + \frac{26962692514090959}{118446184548895651} a^{9} + \frac{35397358381283085}{118446184548895651} a^{8} - \frac{50140596600464448}{118446184548895651} a^{7} - \frac{3273737619056993}{118446184548895651} a^{6} + \frac{3476927333385278}{16920883506985093} a^{5} - \frac{47422317582364615}{118446184548895651} a^{4} - \frac{8043604254500446}{118446184548895651} a^{3} - \frac{6398914844231469}{16920883506985093} a^{2} - \frac{3424858751595686}{16920883506985093} a + \frac{32335407714728763}{118446184548895651}$

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:  \( 170660357100 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T28:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1029
The 133 conjugacy class representatives for t21n28 are not computed
Character table for t21n28 is not computed

Intermediate fields

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

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

Sibling fields

Degree 21 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 $21$ $21$ $21$ R $21$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ $21$ $21$ R $21$ $21$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{7}$ $21$ $21$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
7Data not computed
$29$29.7.0.1$x^{7} - x + 3$$1$$7$$0$$C_7$$[\ ]^{7}$
29.7.6.5$x^{7} + 58$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.0.1$x^{7} - x + 3$$1$$7$$0$$C_7$$[\ ]^{7}$