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Properties

Label 21.21.1427369861...3689.1
Degree $21$
Signature $[21, 0]$
Discriminant $7^{14}\cdot 29^{18}$
Root discriminant $65.60$
Ramified primes $7, 29$
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![2843, -16800, 10664, 97061, -141878, -198823, 401883, 180953, -524134, -59599, 362514, -17833, -138787, 19392, 29504, -5652, -3392, 752, 192, -46, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 - 46*x^19 + 192*x^18 + 752*x^17 - 3392*x^16 - 5652*x^15 + 29504*x^14 + 19392*x^13 - 138787*x^12 - 17833*x^11 + 362514*x^10 - 59599*x^9 - 524134*x^8 + 180953*x^7 + 401883*x^6 - 198823*x^5 - 141878*x^4 + 97061*x^3 + 10664*x^2 - 16800*x + 2843)
 
gp: K = bnfinit(x^21 - 4*x^20 - 46*x^19 + 192*x^18 + 752*x^17 - 3392*x^16 - 5652*x^15 + 29504*x^14 + 19392*x^13 - 138787*x^12 - 17833*x^11 + 362514*x^10 - 59599*x^9 - 524134*x^8 + 180953*x^7 + 401883*x^6 - 198823*x^5 - 141878*x^4 + 97061*x^3 + 10664*x^2 - 16800*x + 2843, 1)
 

Normalized defining polynomial

\( x^{21} - 4 x^{20} - 46 x^{19} + 192 x^{18} + 752 x^{17} - 3392 x^{16} - 5652 x^{15} + 29504 x^{14} + 19392 x^{13} - 138787 x^{12} - 17833 x^{11} + 362514 x^{10} - 59599 x^{9} - 524134 x^{8} + 180953 x^{7} + 401883 x^{6} - 198823 x^{5} - 141878 x^{4} + 97061 x^{3} + 10664 x^{2} - 16800 x + 2843 \)

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:  \(142736986105602839685204351151303673689=7^{14}\cdot 29^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.60$
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 Galois and abelian over $\Q$.
Conductor:  \(203=7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{203}(1,·)$, $\chi_{203}(197,·)$, $\chi_{203}(198,·)$, $\chi_{203}(65,·)$, $\chi_{203}(74,·)$, $\chi_{203}(141,·)$, $\chi_{203}(78,·)$, $\chi_{203}(16,·)$, $\chi_{203}(81,·)$, $\chi_{203}(23,·)$, $\chi_{203}(88,·)$, $\chi_{203}(25,·)$, $\chi_{203}(30,·)$, $\chi_{203}(36,·)$, $\chi_{203}(165,·)$, $\chi_{203}(169,·)$, $\chi_{203}(170,·)$, $\chi_{203}(107,·)$, $\chi_{203}(53,·)$, $\chi_{203}(123,·)$, $\chi_{203}(190,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17} a^{18} + \frac{5}{17} a^{17} - \frac{3}{17} a^{16} + \frac{6}{17} a^{15} - \frac{1}{17} a^{14} - \frac{8}{17} a^{13} - \frac{3}{17} a^{12} - \frac{6}{17} a^{11} + \frac{1}{17} a^{9} + \frac{2}{17} a^{8} + \frac{5}{17} a^{7} - \frac{3}{17} a^{6} - \frac{2}{17} a^{5} - \frac{4}{17} a^{4} - \frac{7}{17} a^{3} - \frac{3}{17} a^{2} - \frac{8}{17} a - \frac{1}{17}$, $\frac{1}{697} a^{19} - \frac{20}{697} a^{18} - \frac{332}{697} a^{17} - \frac{293}{697} a^{16} - \frac{49}{697} a^{15} + \frac{5}{41} a^{14} + \frac{214}{697} a^{13} + \frac{154}{697} a^{12} - \frac{88}{697} a^{11} + \frac{86}{697} a^{10} + \frac{130}{697} a^{9} + \frac{244}{697} a^{8} + \frac{110}{697} a^{7} - \frac{46}{697} a^{6} - \frac{345}{697} a^{5} - \frac{264}{697} a^{4} + \frac{121}{697} a^{3} + \frac{16}{697} a^{2} - \frac{260}{697} a + \frac{331}{697}$, $\frac{1}{94681350923027856166972143748841} a^{20} + \frac{26893875581492066769715027169}{94681350923027856166972143748841} a^{19} + \frac{973746948699479043246103055526}{94681350923027856166972143748841} a^{18} + \frac{31313757912839943211074043291513}{94681350923027856166972143748841} a^{17} - \frac{10295245055016461279992786512442}{94681350923027856166972143748841} a^{16} - \frac{30644002701270934383486487138270}{94681350923027856166972143748841} a^{15} + \frac{421402792740371757653653067699}{2309301242025069662609076676801} a^{14} + \frac{40103743480146765987051837486268}{94681350923027856166972143748841} a^{13} - \frac{14616209773148586652243710363536}{94681350923027856166972143748841} a^{12} + \frac{43869311495837625156551496077736}{94681350923027856166972143748841} a^{11} - \frac{22227426304848921818124577625217}{94681350923027856166972143748841} a^{10} - \frac{14611160237634753382110627388151}{94681350923027856166972143748841} a^{9} - \frac{40947155305647503019542413186887}{94681350923027856166972143748841} a^{8} - \frac{17742929640627350574440022836395}{94681350923027856166972143748841} a^{7} - \frac{22916316161008780830154156552080}{94681350923027856166972143748841} a^{6} - \frac{36170850360752235352617873740474}{94681350923027856166972143748841} a^{5} + \frac{33318582483221936265509382788098}{94681350923027856166972143748841} a^{4} - \frac{619995942956309404930601310542}{94681350923027856166972143748841} a^{3} - \frac{4179560882355591846095379937990}{94681350923027856166972143748841} a^{2} - \frac{17572846455567989954960618687350}{94681350923027856166972143748841} a - \frac{16294892671649717002326733849869}{94681350923027856166972143748841}$

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:  \( 2231765965710 \) (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

\(\Q(\zeta_{7})^+\), 7.7.594823321.1

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 $21$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ $21$ $21$ R $21$ $21$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{21}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ $21$ $21$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$