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Properties

Label 21.21.2118750674...7721.1
Degree $21$
Signature $[21, 0]$
Discriminant $7^{36}\cdot 19^{14}$
Root discriminant $200.10$
Ramified primes $7, 19$
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![-26753, 131824, 562184, -1775109, -4948475, 6621433, 18273234, -4195819, -20496567, 976654, 10297315, -497623, -2582937, 232456, 319090, -37646, -19901, 2639, 602, -84, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 84*x^19 + 602*x^18 + 2639*x^17 - 19901*x^16 - 37646*x^15 + 319090*x^14 + 232456*x^13 - 2582937*x^12 - 497623*x^11 + 10297315*x^10 + 976654*x^9 - 20496567*x^8 - 4195819*x^7 + 18273234*x^6 + 6621433*x^5 - 4948475*x^4 - 1775109*x^3 + 562184*x^2 + 131824*x - 26753)
 
gp: K = bnfinit(x^21 - 7*x^20 - 84*x^19 + 602*x^18 + 2639*x^17 - 19901*x^16 - 37646*x^15 + 319090*x^14 + 232456*x^13 - 2582937*x^12 - 497623*x^11 + 10297315*x^10 + 976654*x^9 - 20496567*x^8 - 4195819*x^7 + 18273234*x^6 + 6621433*x^5 - 4948475*x^4 - 1775109*x^3 + 562184*x^2 + 131824*x - 26753, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 84 x^{19} + 602 x^{18} + 2639 x^{17} - 19901 x^{16} - 37646 x^{15} + 319090 x^{14} + 232456 x^{13} - 2582937 x^{12} - 497623 x^{11} + 10297315 x^{10} + 976654 x^{9} - 20496567 x^{8} - 4195819 x^{7} + 18273234 x^{6} + 6621433 x^{5} - 4948475 x^{4} - 1775109 x^{3} + 562184 x^{2} + 131824 x - 26753 \)

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:  \(2118750674738565668124319822288824029028146457721=7^{36}\cdot 19^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $200.10$
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:  $7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(931=7^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{931}(64,·)$, $\chi_{931}(1,·)$, $\chi_{931}(771,·)$, $\chi_{931}(197,·)$, $\chi_{931}(134,·)$, $\chi_{931}(904,·)$, $\chi_{931}(330,·)$, $\chi_{931}(267,·)$, $\chi_{931}(463,·)$, $\chi_{931}(400,·)$, $\chi_{931}(596,·)$, $\chi_{931}(533,·)$, $\chi_{931}(729,·)$, $\chi_{931}(666,·)$, $\chi_{931}(862,·)$, $\chi_{931}(799,·)$, $\chi_{931}(106,·)$, $\chi_{931}(239,·)$, $\chi_{931}(372,·)$, $\chi_{931}(505,·)$, $\chi_{931}(638,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{31} a^{17} + \frac{1}{31} a^{16} - \frac{10}{31} a^{14} - \frac{15}{31} a^{13} - \frac{6}{31} a^{12} + \frac{7}{31} a^{11} - \frac{12}{31} a^{10} + \frac{12}{31} a^{9} - \frac{6}{31} a^{8} - \frac{1}{31} a^{7} + \frac{7}{31} a^{6} + \frac{13}{31} a^{5} + \frac{5}{31} a^{4} - \frac{11}{31} a^{3} + \frac{6}{31} a^{2} + \frac{10}{31} a$, $\frac{1}{589} a^{18} + \frac{8}{589} a^{17} - \frac{210}{589} a^{16} + \frac{207}{589} a^{15} - \frac{116}{589} a^{14} + \frac{44}{589} a^{13} - \frac{283}{589} a^{12} + \frac{37}{589} a^{11} - \frac{165}{589} a^{10} - \frac{232}{589} a^{9} + \frac{112}{589} a^{8} + \frac{1}{19} a^{7} + \frac{3}{19} a^{6} + \frac{251}{589} a^{5} + \frac{179}{589} a^{4} - \frac{9}{589} a^{3} + \frac{21}{589} a^{2} + \frac{70}{589} a - \frac{1}{19}$, $\frac{1}{11191} a^{19} + \frac{3}{11191} a^{18} + \frac{168}{11191} a^{17} - \frac{92}{11191} a^{16} + \frac{616}{11191} a^{15} - \frac{2967}{11191} a^{14} - \frac{294}{11191} a^{13} + \frac{5423}{11191} a^{12} - \frac{3314}{11191} a^{11} + \frac{2645}{11191} a^{10} + \frac{3343}{11191} a^{9} - \frac{5393}{11191} a^{8} - \frac{3425}{11191} a^{7} - \frac{4945}{11191} a^{6} - \frac{5066}{11191} a^{5} + \frac{1775}{11191} a^{4} + \frac{2536}{11191} a^{3} + \frac{4240}{11191} a^{2} - \frac{2680}{11191} a - \frac{109}{361}$, $\frac{1}{174405010991693265864179185797181523268600684670033930363} a^{20} + \frac{7202909030155519709688083580950871615268339161111121}{174405010991693265864179185797181523268600684670033930363} a^{19} + \frac{142088631322679602859484141135831161207276213543754548}{174405010991693265864179185797181523268600684670033930363} a^{18} + \frac{2100486298513800470409736990700118579076208471428687234}{174405010991693265864179185797181523268600684670033930363} a^{17} - \frac{18105400980629002495254800665898866932176984782667771129}{174405010991693265864179185797181523268600684670033930363} a^{16} + \frac{46339262081816835228069518798869364987792551108268326396}{174405010991693265864179185797181523268600684670033930363} a^{15} + \frac{64368442219939854341271132652876457248701396686720796246}{174405010991693265864179185797181523268600684670033930363} a^{14} - \frac{41864978536210299083621903777516637128055929166569355019}{174405010991693265864179185797181523268600684670033930363} a^{13} - \frac{34455726149818528346332260060981267892299838688229342671}{174405010991693265864179185797181523268600684670033930363} a^{12} - \frac{16378787093509943386432873693404993933572338202192329934}{174405010991693265864179185797181523268600684670033930363} a^{11} - \frac{63188108465683725605718063274708734060692463605899854199}{174405010991693265864179185797181523268600684670033930363} a^{10} - \frac{32601505514292851721572646940191158766821135782980963598}{174405010991693265864179185797181523268600684670033930363} a^{9} - \frac{8712132323429330730412401062226628429461745899509466451}{174405010991693265864179185797181523268600684670033930363} a^{8} + \frac{30761734239375069955898612990645968523563448436975388224}{174405010991693265864179185797181523268600684670033930363} a^{7} - \frac{53626544397245791280176330339964737760937467702821422311}{174405010991693265864179185797181523268600684670033930363} a^{6} + \frac{52268732201754978873183266361998135756447617927362009681}{174405010991693265864179185797181523268600684670033930363} a^{5} + \frac{7123532151622747738643564261932430294340653232695812423}{174405010991693265864179185797181523268600684670033930363} a^{4} - \frac{27312339849981280154755554415619244059553930658066310375}{174405010991693265864179185797181523268600684670033930363} a^{3} + \frac{59500762705096815782907818969666357751511157652016175895}{174405010991693265864179185797181523268600684670033930363} a^{2} + \frac{21640996218440189152584389117901021151203369301429756854}{174405010991693265864179185797181523268600684670033930363} a - \frac{758979162833486355586244204121259317187962359520460381}{5625968096506234382715457606360694298987118860323675173}$

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:  \( 832088128597863300 \) (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

3.3.361.1, 7.7.13841287201.1

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 ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ $21$ $21$ R $21$ $21$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{21}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ $21$ $21$ $21$ $21$ $21$

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$7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
$19$19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$