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Properties

Label 21.21.5767844328...6464.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 7^{14}\cdot 23^{18}$
Root discriminant $97.41$
Ramified primes $2, 7, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_7:C_3$ (as 21T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![189, 33705, -671223, -3302816, 4723953, 16344264, -10678784, -28090537, 8554392, 21127316, -2592179, -7801639, 149067, 1461398, 57349, -135842, -8442, 6077, 339, -127, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 - 127*x^19 + 339*x^18 + 6077*x^17 - 8442*x^16 - 135842*x^15 + 57349*x^14 + 1461398*x^13 + 149067*x^12 - 7801639*x^11 - 2592179*x^10 + 21127316*x^9 + 8554392*x^8 - 28090537*x^7 - 10678784*x^6 + 16344264*x^5 + 4723953*x^4 - 3302816*x^3 - 671223*x^2 + 33705*x + 189)
 
gp: K = bnfinit(x^21 - 4*x^20 - 127*x^19 + 339*x^18 + 6077*x^17 - 8442*x^16 - 135842*x^15 + 57349*x^14 + 1461398*x^13 + 149067*x^12 - 7801639*x^11 - 2592179*x^10 + 21127316*x^9 + 8554392*x^8 - 28090537*x^7 - 10678784*x^6 + 16344264*x^5 + 4723953*x^4 - 3302816*x^3 - 671223*x^2 + 33705*x + 189, 1)
 

Normalized defining polynomial

\( x^{21} - 4 x^{20} - 127 x^{19} + 339 x^{18} + 6077 x^{17} - 8442 x^{16} - 135842 x^{15} + 57349 x^{14} + 1461398 x^{13} + 149067 x^{12} - 7801639 x^{11} - 2592179 x^{10} + 21127316 x^{9} + 8554392 x^{8} - 28090537 x^{7} - 10678784 x^{6} + 16344264 x^{5} + 4723953 x^{4} - 3302816 x^{3} - 671223 x^{2} + 33705 x + 189 \)

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:  \(576784432862339953527413669923017531326464=2^{18}\cdot 7^{14}\cdot 23^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.41$
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:  $2, 7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{3}{7} a^{4} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{7} + \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3}$, $\frac{1}{7} a^{8} - \frac{3}{7} a^{5} + \frac{1}{7} a^{4} + \frac{1}{7} a^{3}$, $\frac{1}{49} a^{9} + \frac{1}{49} a^{8} + \frac{1}{49} a^{7} + \frac{1}{49} a^{6} + \frac{1}{49} a^{5} + \frac{8}{49} a^{4} + \frac{1}{49} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{49} a^{10} + \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{15}{49} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{343} a^{11} - \frac{1}{343} a^{10} + \frac{1}{49} a^{8} + \frac{2}{49} a^{7} - \frac{3}{49} a^{6} - \frac{5}{49} a^{5} - \frac{169}{343} a^{4} - \frac{167}{343} a^{3} - \frac{12}{49} a^{2} + \frac{17}{49} a - \frac{1}{49}$, $\frac{1}{686} a^{12} + \frac{3}{343} a^{10} + \frac{1}{49} a^{8} - \frac{1}{49} a^{7} - \frac{1}{49} a^{6} + \frac{17}{343} a^{5} + \frac{1}{14} a^{4} + \frac{137}{343} a^{3} + \frac{20}{49} a^{2} + \frac{22}{49} a + \frac{13}{98}$, $\frac{1}{686} a^{13} + \frac{3}{343} a^{10} + \frac{2}{49} a^{8} - \frac{1}{49} a^{7} + \frac{24}{343} a^{6} - \frac{1}{2} a^{5} - \frac{3}{7} a^{4} - \frac{101}{343} a^{3} + \frac{23}{49} a^{2} - \frac{5}{98} a + \frac{10}{49}$, $\frac{1}{4802} a^{14} + \frac{3}{4802} a^{13} - \frac{1}{4802} a^{12} - \frac{2}{2401} a^{11} - \frac{24}{2401} a^{10} - \frac{3}{343} a^{9} + \frac{8}{343} a^{8} + \frac{52}{2401} a^{7} + \frac{151}{4802} a^{6} + \frac{981}{4802} a^{5} - \frac{1865}{4802} a^{4} + \frac{262}{2401} a^{3} - \frac{249}{686} a^{2} + \frac{57}{686} a - \frac{139}{686}$, $\frac{1}{4802} a^{15} - \frac{3}{4802} a^{13} - \frac{1}{4802} a^{12} + \frac{3}{2401} a^{11} + \frac{2}{2401} a^{10} + \frac{3}{343} a^{9} + \frac{31}{2401} a^{8} + \frac{19}{686} a^{7} - \frac{107}{2401} a^{6} + \frac{43}{4802} a^{5} + \frac{883}{4802} a^{4} - \frac{865}{4802} a^{3} + \frac{17}{343} a^{2} - \frac{317}{686} a - \frac{73}{686}$, $\frac{1}{14406} a^{16} + \frac{1}{14406} a^{14} - \frac{1}{4802} a^{13} - \frac{5}{14406} a^{12} - \frac{2}{2401} a^{11} + \frac{3}{2401} a^{10} + \frac{15}{2401} a^{9} + \frac{41}{2058} a^{8} + \frac{346}{7203} a^{7} + \frac{269}{14406} a^{6} + \frac{2021}{14406} a^{5} - \frac{40}{2401} a^{4} - \frac{485}{7203} a^{3} + \frac{311}{2058} a^{2} + \frac{211}{2058} a + \frac{181}{686}$, $\frac{1}{100842} a^{17} + \frac{1}{14406} a^{15} - \frac{4}{7203} a^{13} - \frac{3}{4802} a^{12} - \frac{2}{2401} a^{11} + \frac{79}{16807} a^{10} + \frac{59}{14406} a^{9} + \frac{499}{7203} a^{8} + \frac{407}{14406} a^{7} + \frac{25}{7203} a^{6} - \frac{207}{4802} a^{5} + \frac{6095}{14406} a^{4} - \frac{35755}{100842} a^{3} - \frac{2980}{7203} a^{2} + \frac{593}{2401} a + \frac{961}{4802}$, $\frac{1}{100842} a^{18} + \frac{5}{14406} a^{12} + \frac{2}{16807} a^{11} - \frac{55}{14406} a^{10} - \frac{29}{7203} a^{9} - \frac{8}{2401} a^{8} - \frac{3}{49} a^{7} - \frac{11}{294} a^{6} - \frac{842}{2401} a^{5} + \frac{2045}{100842} a^{4} - \frac{1334}{7203} a^{3} + \frac{1373}{7203} a^{2} - \frac{515}{7203} a - \frac{339}{686}$, $\frac{1}{41111972454} a^{19} - \frac{116447}{41111972454} a^{18} - \frac{121547}{41111972454} a^{17} + \frac{27493}{978856487} a^{16} - \frac{2164}{2936569461} a^{15} - \frac{41233}{1957712974} a^{14} - \frac{677323}{978856487} a^{13} + \frac{34952}{49059633} a^{12} - \frac{40961455}{41111972454} a^{11} + \frac{265752163}{41111972454} a^{10} + \frac{18341755}{5873138922} a^{9} - \frac{110185769}{2936569461} a^{8} + \frac{3481577}{1957712974} a^{7} + \frac{36803371}{978856487} a^{6} - \frac{17901150427}{41111972454} a^{5} - \frac{1097507500}{6851995409} a^{4} + \frac{4167808558}{20555986227} a^{3} - \frac{1320267089}{5873138922} a^{2} - \frac{802159076}{2936569461} a + \frac{473643990}{978856487}$, $\frac{1}{248153814601718331401571384594} a^{20} + \frac{566841907244228629}{124076907300859165700785692297} a^{19} - \frac{32155782977487477190487}{124076907300859165700785692297} a^{18} - \frac{6704930360234297391080}{1969474719061256598425169719} a^{17} + \frac{437061861518419340503957}{17725272471551309385826527471} a^{16} + \frac{121579413762028745733065}{5908424157183769795275509157} a^{15} - \frac{2382018650374618034121311}{35450544943102618771653054942} a^{14} - \frac{5301998783577259165537991}{248153814601718331401571384594} a^{13} - \frac{58114163697652323521859821}{124076907300859165700785692297} a^{12} + \frac{35612997938848309119987877}{41358969100286388566928564099} a^{11} + \frac{5482302227432880433897885}{17725272471551309385826527471} a^{10} + \frac{19711740900465324871454363}{17725272471551309385826527471} a^{9} - \frac{1952369093158896299770736977}{35450544943102618771653054942} a^{8} + \frac{107558407399862534777761060}{5908424157183769795275509157} a^{7} + \frac{4743832806386149227735159905}{248153814601718331401571384594} a^{6} + \frac{95333540540974294795494357769}{248153814601718331401571384594} a^{5} - \frac{26669084087508232659520655083}{82717938200572777133857128198} a^{4} + \frac{268491470177385892796197457}{844060593883395685039358451} a^{3} + \frac{6601912332185077965290178607}{35450544943102618771653054942} a^{2} - \frac{889841254335357161967801515}{3938949438122513196850339438} a - \frac{116633002809234129358728123}{562707062588930456692905634}$

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:  \( 434748805809000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7:C_3$ (as 21T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 21
The 5 conjugacy class representatives for $C_7:C_3$
Character table for $C_7:C_3$

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.7.22747786847296.1 x7

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

Sibling fields

Degree 7 sibling: 7.7.22747786847296.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
2Data not computed
$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}$
23Data not computed