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Properties

Label 21.21.1713186962...3001.1
Degree $21$
Signature $[21, 0]$
Discriminant $7^{14}\cdot 43^{18}$
Root discriminant $91.94$
Ramified primes $7, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{21}$ (as 21T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1757, 6678, -71820, -51323, 548114, 381991, -1550169, -1520495, 1352446, 1805255, -320638, -844197, -27551, 195304, 21110, -24588, -3096, 1724, 186, -64, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 - 64*x^19 + 186*x^18 + 1724*x^17 - 3096*x^16 - 24588*x^15 + 21110*x^14 + 195304*x^13 - 27551*x^12 - 844197*x^11 - 320638*x^10 + 1805255*x^9 + 1352446*x^8 - 1520495*x^7 - 1550169*x^6 + 381991*x^5 + 548114*x^4 - 51323*x^3 - 71820*x^2 + 6678*x + 1757)
 
gp: K = bnfinit(x^21 - 4*x^20 - 64*x^19 + 186*x^18 + 1724*x^17 - 3096*x^16 - 24588*x^15 + 21110*x^14 + 195304*x^13 - 27551*x^12 - 844197*x^11 - 320638*x^10 + 1805255*x^9 + 1352446*x^8 - 1520495*x^7 - 1550169*x^6 + 381991*x^5 + 548114*x^4 - 51323*x^3 - 71820*x^2 + 6678*x + 1757, 1)
 

Normalized defining polynomial

\( x^{21} - 4 x^{20} - 64 x^{19} + 186 x^{18} + 1724 x^{17} - 3096 x^{16} - 24588 x^{15} + 21110 x^{14} + 195304 x^{13} - 27551 x^{12} - 844197 x^{11} - 320638 x^{10} + 1805255 x^{9} + 1352446 x^{8} - 1520495 x^{7} - 1550169 x^{6} + 381991 x^{5} + 548114 x^{4} - 51323 x^{3} - 71820 x^{2} + 6678 x + 1757 \)

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:  \(171318696215827426793735775028238670573001=7^{14}\cdot 43^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.94$
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:  $7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(301=7\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{301}(256,·)$, $\chi_{301}(1,·)$, $\chi_{301}(130,·)$, $\chi_{301}(4,·)$, $\chi_{301}(193,·)$, $\chi_{301}(64,·)$, $\chi_{301}(11,·)$, $\chi_{301}(78,·)$, $\chi_{301}(207,·)$, $\chi_{301}(16,·)$, $\chi_{301}(274,·)$, $\chi_{301}(219,·)$, $\chi_{301}(226,·)$, $\chi_{301}(102,·)$, $\chi_{301}(170,·)$, $\chi_{301}(107,·)$, $\chi_{301}(44,·)$, $\chi_{301}(176,·)$, $\chi_{301}(183,·)$, $\chi_{301}(121,·)$, $\chi_{301}(127,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{14} - \frac{1}{7} a^{13} + \frac{1}{7} a^{12} + \frac{2}{7} a^{11} - \frac{1}{7} a^{10} + \frac{3}{7} a^{9} + \frac{1}{7} a^{8} + \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{16} - \frac{2}{7} a^{14} + \frac{2}{7} a^{13} + \frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{3}{7} a^{10} - \frac{2}{7} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3}$, $\frac{1}{49} a^{17} - \frac{3}{49} a^{16} - \frac{3}{49} a^{15} - \frac{3}{7} a^{14} - \frac{4}{49} a^{13} + \frac{2}{7} a^{12} + \frac{4}{49} a^{11} + \frac{1}{49} a^{10} + \frac{4}{49} a^{9} + \frac{3}{7} a^{7} - \frac{20}{49} a^{6} - \frac{4}{49} a^{5} - \frac{20}{49} a^{4} - \frac{19}{49} a^{3} - \frac{2}{7} a^{2} + \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{49} a^{18} + \frac{2}{49} a^{16} - \frac{2}{49} a^{15} - \frac{18}{49} a^{14} + \frac{2}{49} a^{13} - \frac{10}{49} a^{12} - \frac{22}{49} a^{11} - \frac{2}{7} a^{10} + \frac{19}{49} a^{9} + \frac{2}{7} a^{8} + \frac{8}{49} a^{7} + \frac{6}{49} a^{6} - \frac{11}{49} a^{5} + \frac{19}{49} a^{4} - \frac{8}{49} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7}$, $\frac{1}{49} a^{19} - \frac{3}{49} a^{16} + \frac{2}{49} a^{15} + \frac{23}{49} a^{14} + \frac{19}{49} a^{13} + \frac{6}{49} a^{12} - \frac{22}{49} a^{11} + \frac{24}{49} a^{10} + \frac{13}{49} a^{9} + \frac{15}{49} a^{8} + \frac{13}{49} a^{7} + \frac{15}{49} a^{6} + \frac{13}{49} a^{5} + \frac{4}{49} a^{4} - \frac{11}{49} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{48554480969669843311746601589140852912281833} a^{20} + \frac{437819134590870011939769998268536781506339}{48554480969669843311746601589140852912281833} a^{19} + \frac{355339073439566147293731951457149580587173}{48554480969669843311746601589140852912281833} a^{18} - \frac{11010451790449536066461298476143775969756}{6936354424238549044535228798448693273183119} a^{17} + \frac{521830581650921296921666010090385341148456}{48554480969669843311746601589140852912281833} a^{16} + \frac{264007482905892794618976509812502732584453}{6936354424238549044535228798448693273183119} a^{15} - \frac{5348314293406639043932414341839307677700543}{48554480969669843311746601589140852912281833} a^{14} - \frac{4204626640681563991000421501165907655595153}{48554480969669843311746601589140852912281833} a^{13} + \frac{16894200777192460387025919753887545935010212}{48554480969669843311746601589140852912281833} a^{12} - \frac{349033122319294916768591753976738671070330}{990907774891221292076461256921241896169017} a^{11} + \frac{2711815990851674930382293716919258280574972}{6936354424238549044535228798448693273183119} a^{10} - \frac{19957117135776963360563341262027579752291860}{48554480969669843311746601589140852912281833} a^{9} - \frac{23364394194418364092558747173697329149033262}{48554480969669843311746601589140852912281833} a^{8} + \frac{10312819118782822591786446906637783219634837}{48554480969669843311746601589140852912281833} a^{7} + \frac{2614915781449557312610780389409629702857702}{48554480969669843311746601589140852912281833} a^{6} + \frac{1876979557602360964090963131439057992131493}{6936354424238549044535228798448693273183119} a^{5} - \frac{2562811416364194060742890974299456270835274}{6936354424238549044535228798448693273183119} a^{4} - \frac{1183251966773673731311515794756204475612530}{6936354424238549044535228798448693273183119} a^{3} + \frac{104837300059569930578896771348990092628889}{990907774891221292076461256921241896169017} a^{2} - \frac{267084280029313463552296289690441078737330}{990907774891221292076461256921241896169017} a - \frac{230762981945737718488778666666581802562838}{990907774891221292076461256921241896169017}$

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:  \( 140330847878000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{21}$ (as 21T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 21
The 21 conjugacy class representatives for $C_{21}$
Character table for $C_{21}$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.7.6321363049.1

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

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 $21$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ $21$ $21$ $21$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ R $21$ $21$ $21$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$43$43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$