↑

Properties

Label 21.21.1351935538...9824.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 7^{17}\cdot 43^{3}\cdot 127^{3}\cdot 449^{3}\cdot 24683^{3}$
Root discriminant $303.69$
Ramified primes $2, 7, 43, 127, 449, 24683$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T45

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![33368, -562728, 3544684, -9898284, 8410376, 15488942, -34903053, 9469379, 25975820, -20222740, -3181310, 8184116, -1915634, -943874, 484726, -14628, -27416, 3748, 534, -112, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 112*x^19 + 534*x^18 + 3748*x^17 - 27416*x^16 - 14628*x^15 + 484726*x^14 - 943874*x^13 - 1915634*x^12 + 8184116*x^11 - 3181310*x^10 - 20222740*x^9 + 25975820*x^8 + 9469379*x^7 - 34903053*x^6 + 15488942*x^5 + 8410376*x^4 - 9898284*x^3 + 3544684*x^2 - 562728*x + 33368)
 
gp: K = bnfinit(x^21 - 3*x^20 - 112*x^19 + 534*x^18 + 3748*x^17 - 27416*x^16 - 14628*x^15 + 484726*x^14 - 943874*x^13 - 1915634*x^12 + 8184116*x^11 - 3181310*x^10 - 20222740*x^9 + 25975820*x^8 + 9469379*x^7 - 34903053*x^6 + 15488942*x^5 + 8410376*x^4 - 9898284*x^3 + 3544684*x^2 - 562728*x + 33368, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} - 112 x^{19} + 534 x^{18} + 3748 x^{17} - 27416 x^{16} - 14628 x^{15} + 484726 x^{14} - 943874 x^{13} - 1915634 x^{12} + 8184116 x^{11} - 3181310 x^{10} - 20222740 x^{9} + 25975820 x^{8} + 9469379 x^{7} - 34903053 x^{6} + 15488942 x^{5} + 8410376 x^{4} - 9898284 x^{3} + 3544684 x^{2} - 562728 x + 33368 \)

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:  \(13519355382235439206682551551249436467623411507789824=2^{18}\cdot 7^{17}\cdot 43^{3}\cdot 127^{3}\cdot 449^{3}\cdot 24683^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $303.69$
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:  $2, 7, 43, 127, 449, 24683$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{1}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{8} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a^{2} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{9} + \frac{3}{7} a^{6} + \frac{3}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{11} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a^{2} - \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{49} a^{12} - \frac{2}{49} a^{10} + \frac{3}{49} a^{9} + \frac{2}{49} a^{8} + \frac{3}{49} a^{7} - \frac{3}{7} a^{6} - \frac{22}{49} a^{5} + \frac{2}{7} a^{4} - \frac{12}{49} a^{3} - \frac{10}{49} a^{2} - \frac{16}{49} a + \frac{11}{49}$, $\frac{1}{49} a^{13} - \frac{2}{49} a^{11} + \frac{3}{49} a^{10} + \frac{2}{49} a^{9} + \frac{3}{49} a^{8} + \frac{6}{49} a^{6} - \frac{1}{7} a^{5} + \frac{16}{49} a^{4} + \frac{18}{49} a^{3} + \frac{12}{49} a^{2} - \frac{10}{49} a + \frac{1}{7}$, $\frac{1}{49} a^{14} + \frac{3}{49} a^{11} - \frac{2}{49} a^{10} + \frac{2}{49} a^{9} - \frac{3}{49} a^{8} - \frac{2}{49} a^{7} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{4}{49} a^{4} - \frac{5}{49} a^{3} - \frac{16}{49} a^{2} - \frac{11}{49} a - \frac{6}{49}$, $\frac{1}{98} a^{15} - \frac{1}{98} a^{14} - \frac{1}{98} a^{13} - \frac{1}{98} a^{12} - \frac{3}{98} a^{11} - \frac{5}{98} a^{10} - \frac{5}{98} a^{9} - \frac{3}{98} a^{8} - \frac{3}{98} a^{7} + \frac{43}{98} a^{6} + \frac{15}{98} a^{5} - \frac{39}{98} a^{4} - \frac{23}{98} a^{3} - \frac{37}{98} a^{2} + \frac{8}{49} a + \frac{23}{49}$, $\frac{1}{98} a^{16} - \frac{3}{49} a^{11} - \frac{1}{49} a^{10} - \frac{1}{49} a^{9} + \frac{1}{49} a^{8} + \frac{3}{49} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} + \frac{10}{49} a^{4} - \frac{6}{49} a^{3} + \frac{29}{98} a^{2} + \frac{20}{49} a + \frac{11}{49}$, $\frac{1}{1372} a^{17} + \frac{1}{1372} a^{16} + \frac{5}{1372} a^{15} + \frac{13}{1372} a^{14} - \frac{1}{196} a^{13} - \frac{1}{196} a^{12} + \frac{11}{196} a^{11} - \frac{65}{1372} a^{10} + \frac{19}{1372} a^{9} + \frac{81}{1372} a^{8} - \frac{89}{1372} a^{7} - \frac{83}{196} a^{6} - \frac{27}{196} a^{5} + \frac{17}{196} a^{4} - \frac{115}{686} a^{3} + \frac{333}{686} a^{2} + \frac{101}{343} a + \frac{117}{343}$, $\frac{1}{1372} a^{18} + \frac{1}{343} a^{16} - \frac{3}{686} a^{15} - \frac{3}{686} a^{14} - \frac{1}{98} a^{13} - \frac{1}{98} a^{12} - \frac{11}{343} a^{11} - \frac{1}{14} a^{10} - \frac{16}{343} a^{9} - \frac{11}{343} a^{8} - \frac{1}{686} a^{7} + \frac{1}{98} a^{6} - \frac{41}{98} a^{5} + \frac{337}{1372} a^{4} + \frac{5}{14} a^{3} + \frac{64}{343} a^{2} - \frac{110}{343} a + \frac{51}{343}$, $\frac{1}{74088} a^{19} + \frac{5}{74088} a^{18} - \frac{1}{10584} a^{17} + \frac{101}{74088} a^{16} + \frac{5}{3528} a^{15} - \frac{355}{74088} a^{14} + \frac{1}{1512} a^{13} - \frac{737}{74088} a^{12} + \frac{3707}{74088} a^{11} - \frac{545}{10584} a^{10} - \frac{937}{74088} a^{9} - \frac{263}{10584} a^{8} - \frac{4309}{74088} a^{7} - \frac{565}{1512} a^{6} - \frac{955}{37044} a^{5} - \frac{1747}{4116} a^{4} - \frac{11}{54} a^{3} - \frac{5573}{18522} a^{2} + \frac{17}{1323} a + \frac{583}{9261}$, $\frac{1}{11687869933072140657752365990427712} a^{20} - \frac{12361056301422525497103158879}{2921967483268035164438091497606928} a^{19} + \frac{11526831513422409817494744889}{108221017898816117201410796207664} a^{18} - \frac{69538760887077652421116506889}{1947978322178690109625394331737952} a^{17} - \frac{577596126418847663710083421771}{5843934966536070328876182995213856} a^{16} + \frac{28777259676788892142074593406823}{5843934966536070328876182995213856} a^{15} - \frac{58947877874558749387785641900225}{5843934966536070328876182995213856} a^{14} - \frac{1877277893628068158651192103737}{486994580544672527406348582934488} a^{13} - \frac{17445311415536246016805440957301}{5843934966536070328876182995213856} a^{12} - \frac{48450370409588316215003016445331}{1460983741634017582219045748803464} a^{11} + \frac{205543452559712815941262585000147}{2921967483268035164438091497606928} a^{10} - \frac{116866589101989865807790252933717}{5843934966536070328876182995213856} a^{9} - \frac{60575699938707802191506736280093}{5843934966536070328876182995213856} a^{8} - \frac{13530275235310336423541156860799}{216442035797632234402821592415328} a^{7} - \frac{124591714372135322061398339005699}{11687869933072140657752365990427712} a^{6} + \frac{2659444427271085394663400739021327}{5843934966536070328876182995213856} a^{5} + \frac{86717729913521006703662402585933}{365245935408504395554761437200866} a^{4} + \frac{37136551077606950005185615524227}{486994580544672527406348582934488} a^{3} - \frac{94605699954710313503911517238095}{973989161089345054812697165868976} a^{2} + \frac{52073860199820896414161526944519}{182622967704252197777380718600433} a + \frac{531187358270029839232948656693971}{1460983741634017582219045748803464}$

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:  \( 101170607775000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T45:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8232
The 55 conjugacy class representatives for t21n45 are not computed
Character table for t21n45 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 28 siblings: 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{7}$ R $21$ ${\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $21$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.7.0.1$x^{7} - 2 x + 9$$1$$7$$0$$C_7$$[\ ]^{7}$
127Data not computed
449Data not computed
24683Data not computed