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Properties

Label 21.21.2986476193...7728.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 7^{29}\cdot 29^{12}$
Root discriminant $182.27$
Ramified primes $2, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_7:(C_3\times D_7)$ (as 21T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![364464491, 1114112720, -2362982118, -3704545313, 9329430684, -4087297767, -3053607823, 2966333481, -174494054, -561540210, 146916245, 43800414, -20451844, -1028629, 1363504, -58303, -49049, 4501, 917, -112, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 112*x^19 + 917*x^18 + 4501*x^17 - 49049*x^16 - 58303*x^15 + 1363504*x^14 - 1028629*x^13 - 20451844*x^12 + 43800414*x^11 + 146916245*x^10 - 561540210*x^9 - 174494054*x^8 + 2966333481*x^7 - 3053607823*x^6 - 4087297767*x^5 + 9329430684*x^4 - 3704545313*x^3 - 2362982118*x^2 + 1114112720*x + 364464491)
 
gp: K = bnfinit(x^21 - 7*x^20 - 112*x^19 + 917*x^18 + 4501*x^17 - 49049*x^16 - 58303*x^15 + 1363504*x^14 - 1028629*x^13 - 20451844*x^12 + 43800414*x^11 + 146916245*x^10 - 561540210*x^9 - 174494054*x^8 + 2966333481*x^7 - 3053607823*x^6 - 4087297767*x^5 + 9329430684*x^4 - 3704545313*x^3 - 2362982118*x^2 + 1114112720*x + 364464491, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 112 x^{19} + 917 x^{18} + 4501 x^{17} - 49049 x^{16} - 58303 x^{15} + 1363504 x^{14} - 1028629 x^{13} - 20451844 x^{12} + 43800414 x^{11} + 146916245 x^{10} - 561540210 x^{9} - 174494054 x^{8} + 2966333481 x^{7} - 3053607823 x^{6} - 4087297767 x^{5} + 9329430684 x^{4} - 3704545313 x^{3} - 2362982118 x^{2} + 1114112720 x + 364464491 \)

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:  \(298647619353990167618744853614623738567035977728=2^{18}\cdot 7^{29}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.27$
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:  $2, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{14} - \frac{1}{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{14} + \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{3}{8} a^{7} + \frac{3}{8} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} - \frac{3}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{14} + \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{3}{8} a^{6} - \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} - \frac{3}{8} a - \frac{1}{8}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{13} + \frac{1}{8} a^{12} + \frac{1}{8} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} + \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{3}{8} a + \frac{3}{8}$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{14} + \frac{1}{8} a^{13} + \frac{1}{8} a^{10} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{3}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} + \frac{3}{8} a$, $\frac{1}{8} a^{19} + \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{3}{8} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{7473074414800798133831027079099597125561499350665277127522498232} a^{20} - \frac{39942389418623799374010041321071163418918912796468933194058437}{934134301850099766728878384887449640695187418833159640940312279} a^{19} + \frac{18679035070973680741200382406591316269829725690824987917037187}{934134301850099766728878384887449640695187418833159640940312279} a^{18} + \frac{272392293929710502433529716358902121981482715503743478070064847}{7473074414800798133831027079099597125561499350665277127522498232} a^{17} - \frac{2684318522021243835411284347922046203626491709373272669346051}{934134301850099766728878384887449640695187418833159640940312279} a^{16} - \frac{29983655281710412919277012743931090338761587349120580603709610}{934134301850099766728878384887449640695187418833159640940312279} a^{15} + \frac{1662201814092754502161738050397527666554694706675404730951356}{934134301850099766728878384887449640695187418833159640940312279} a^{14} - \frac{439246125659285508588391364131336495599014175172324499887284191}{3736537207400399066915513539549798562780749675332638563761249116} a^{13} + \frac{14016044361998598288975730291528381826783319072226604637717859}{91135053839034123583305208281702403970262187203235086921006076} a^{12} - \frac{911821912029511779695047621222884195673384752249787271663076811}{3736537207400399066915513539549798562780749675332638563761249116} a^{11} - \frac{584437263401493443236275107698215862936672998053333451534958207}{3736537207400399066915513539549798562780749675332638563761249116} a^{10} - \frac{89948767310661142462921194371014117551328339673791590922507267}{7473074414800798133831027079099597125561499350665277127522498232} a^{9} + \frac{1085923122161889011046578368155602456287802786060233872075105425}{7473074414800798133831027079099597125561499350665277127522498232} a^{8} + \frac{726289704864260155056782477342630902473349258017930279067586175}{1868268603700199533457756769774899281390374837666319281880624558} a^{7} - \frac{636740622254079879310327032526697185372770066608363732741724725}{1868268603700199533457756769774899281390374837666319281880624558} a^{6} - \frac{1330031359846305238164546039693704769618965784425776142563996247}{7473074414800798133831027079099597125561499350665277127522498232} a^{5} - \frac{2163751828381890053866373997117737115151489604715705155908849741}{7473074414800798133831027079099597125561499350665277127522498232} a^{4} + \frac{553803645152154748084821335105363467444242207830680834889435205}{1868268603700199533457756769774899281390374837666319281880624558} a^{3} - \frac{22095173491671107522678760936520313183003465270059582314017959}{45567526919517061791652604140851201985131093601617543460503038} a^{2} + \frac{20785974436805575808229678964709878432196593422025598264265493}{287425939030799928224270272273061427906211513487126043366249932} a + \frac{3194741896866683172440820722249210176651027082107053844588024517}{7473074414800798133831027079099597125561499350665277127522498232}$

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:  \( 238931036050000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7:(C_3\times D_7)$ (as 21T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 294
The 22 conjugacy class representatives for $C_7:(C_3\times D_7)$
Character table for $C_7:(C_3\times D_7)$ 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 sibling: data not computed
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 $21$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ R $21$ $21$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ $21$ $21$

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$$[\ ]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
7Data not computed
$29$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.7$x^{7} - 116$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.0.1$x^{7} - x + 3$$1$$7$$0$$C_7$$[\ ]^{7}$