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Properties

Label 21.21.3225600451...2576.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 7^{14}\cdot 1621^{6}$
Root discriminant $54.77$
Ramified primes $2, 7, 1621$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T44

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-113, 9946, 26935, -106383, -276738, 197792, 593915, -157934, -587331, 67407, 327519, -18090, -111319, 4211, 23585, -1066, -3052, 211, 222, -23, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 23*x^19 + 222*x^18 + 211*x^17 - 3052*x^16 - 1066*x^15 + 23585*x^14 + 4211*x^13 - 111319*x^12 - 18090*x^11 + 327519*x^10 + 67407*x^9 - 587331*x^8 - 157934*x^7 + 593915*x^6 + 197792*x^5 - 276738*x^4 - 106383*x^3 + 26935*x^2 + 9946*x - 113)
 
gp: K = bnfinit(x^21 - 7*x^20 - 23*x^19 + 222*x^18 + 211*x^17 - 3052*x^16 - 1066*x^15 + 23585*x^14 + 4211*x^13 - 111319*x^12 - 18090*x^11 + 327519*x^10 + 67407*x^9 - 587331*x^8 - 157934*x^7 + 593915*x^6 + 197792*x^5 - 276738*x^4 - 106383*x^3 + 26935*x^2 + 9946*x - 113, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 23 x^{19} + 222 x^{18} + 211 x^{17} - 3052 x^{16} - 1066 x^{15} + 23585 x^{14} + 4211 x^{13} - 111319 x^{12} - 18090 x^{11} + 327519 x^{10} + 67407 x^{9} - 587331 x^{8} - 157934 x^{7} + 593915 x^{6} + 197792 x^{5} - 276738 x^{4} - 106383 x^{3} + 26935 x^{2} + 9946 x - 113 \)

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:  \(3225600451670981203653263815902232576=2^{18}\cdot 7^{14}\cdot 1621^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.77$
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:  $2, 7, 1621$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{28} a^{18} + \frac{1}{28} a^{17} + \frac{1}{28} a^{16} + \frac{1}{14} a^{14} + \frac{1}{28} a^{13} - \frac{11}{28} a^{12} - \frac{9}{28} a^{11} + \frac{1}{4} a^{10} - \frac{3}{7} a^{9} - \frac{5}{28} a^{8} + \frac{1}{14} a^{7} + \frac{2}{7} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{28} a^{3} + \frac{3}{14} a^{2} - \frac{3}{28} a + \frac{3}{14}$, $\frac{1}{28} a^{19} - \frac{1}{28} a^{16} + \frac{1}{14} a^{15} - \frac{1}{28} a^{14} + \frac{1}{14} a^{13} - \frac{3}{7} a^{12} + \frac{1}{14} a^{11} + \frac{9}{28} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{3}{14} a^{7} - \frac{1}{28} a^{6} + \frac{3}{14} a^{4} - \frac{1}{4} a^{3} + \frac{5}{28} a^{2} + \frac{9}{28} a + \frac{2}{7}$, $\frac{1}{1331832848343918157599964} a^{20} - \frac{376943255778310051259}{1331832848343918157599964} a^{19} + \frac{1495777668242590596756}{332958212085979539399991} a^{18} - \frac{32058677317974404642299}{1331832848343918157599964} a^{17} + \frac{299935360101044349610879}{1331832848343918157599964} a^{16} + \frac{205516010784980090236519}{1331832848343918157599964} a^{15} + \frac{265824753838460467582331}{1331832848343918157599964} a^{14} + \frac{16566540835920758417319}{95130917738851296971426} a^{13} + \frac{161187415629707912436125}{332958212085979539399991} a^{12} - \frac{83464480979351620554097}{1331832848343918157599964} a^{11} + \frac{94391013724308153167971}{665916424171959078799982} a^{10} + \frac{120730644238664580890435}{332958212085979539399991} a^{9} - \frac{130489017656534853702831}{1331832848343918157599964} a^{8} - \frac{179049994439815099746707}{1331832848343918157599964} a^{7} - \frac{382672081413794973821639}{1331832848343918157599964} a^{6} - \frac{34436123994906019182213}{665916424171959078799982} a^{5} - \frac{605280620775269495598757}{1331832848343918157599964} a^{4} - \frac{55251255248567448115935}{332958212085979539399991} a^{3} - \frac{14336777479172263388551}{665916424171959078799982} a^{2} + \frac{105566995221031529089287}{1331832848343918157599964} a - \frac{16436426948081230751578}{332958212085979539399991}$

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:  \( 387606904916 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T44:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7560
The 27 conjugacy class representatives for t21n44
Character table for t21n44 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.7.8240282176.1

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

Sibling fields

Degree 45 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 $21$ $21$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ $21$ $21$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ $21$ $15{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
$2$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.12.12.33$x^{12} + 6 x^{11} - 4 x^{9} - 2 x^{8} + 8 x^{7} + 8 x^{6} - 4 x^{5} + 8 x^{3} + 8 x^{2} + 8$$4$$3$$12$12T45$[4/3, 4/3]_{3}^{6}$
$7$7.6.4.3$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}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
1621Data not computed