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Properties

Label 21.21.3824733046...2304.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 7^{17}\cdot 13^{3}\cdot 305624493259^{3}$
Root discriminant $552.11$
Ramified primes $2, 7, 13, 305624493259$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T45

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9903431, -68881097, 57261767, 382188985, -393645138, -811949642, 436774783, 569575429, -268059978, -175933680, 93151164, 22982466, -17045238, -392050, 1481799, -141881, -51069, 8067, 702, -154, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 154*x^19 + 702*x^18 + 8067*x^17 - 51069*x^16 - 141881*x^15 + 1481799*x^14 - 392050*x^13 - 17045238*x^12 + 22982466*x^11 + 93151164*x^10 - 175933680*x^9 - 268059978*x^8 + 569575429*x^7 + 436774783*x^6 - 811949642*x^5 - 393645138*x^4 + 382188985*x^3 + 57261767*x^2 - 68881097*x + 9903431)
 
gp: K = bnfinit(x^21 - 3*x^20 - 154*x^19 + 702*x^18 + 8067*x^17 - 51069*x^16 - 141881*x^15 + 1481799*x^14 - 392050*x^13 - 17045238*x^12 + 22982466*x^11 + 93151164*x^10 - 175933680*x^9 - 268059978*x^8 + 569575429*x^7 + 436774783*x^6 - 811949642*x^5 - 393645138*x^4 + 382188985*x^3 + 57261767*x^2 - 68881097*x + 9903431, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} - 154 x^{19} + 702 x^{18} + 8067 x^{17} - 51069 x^{16} - 141881 x^{15} + 1481799 x^{14} - 392050 x^{13} - 17045238 x^{12} + 22982466 x^{11} + 93151164 x^{10} - 175933680 x^{9} - 268059978 x^{8} + 569575429 x^{7} + 436774783 x^{6} - 811949642 x^{5} - 393645138 x^{4} + 382188985 x^{3} + 57261767 x^{2} - 68881097 x + 9903431 \)

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:  \(3824733046663878899659290908522222609035545322716565602304=2^{18}\cdot 7^{17}\cdot 13^{3}\cdot 305624493259^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $552.11$
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:  $2, 7, 13, 305624493259$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{1}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{8} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a^{2} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{9} + \frac{3}{7} a^{6} + \frac{3}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{11} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a^{2} - \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{49} a^{12} - \frac{2}{49} a^{10} + \frac{3}{49} a^{9} + \frac{2}{49} a^{8} + \frac{3}{49} a^{7} - \frac{3}{7} a^{6} + \frac{13}{49} a^{5} - \frac{12}{49} a^{3} + \frac{18}{49} a^{2} + \frac{5}{49} a - \frac{10}{49}$, $\frac{1}{294} a^{13} - \frac{1}{294} a^{12} - \frac{3}{98} a^{11} + \frac{5}{294} a^{10} - \frac{1}{294} a^{9} + \frac{5}{98} a^{8} - \frac{1}{98} a^{7} - \frac{33}{98} a^{6} + \frac{33}{98} a^{5} + \frac{107}{294} a^{4} - \frac{103}{294} a^{3} + \frac{33}{98} a^{2} + \frac{83}{294} a + \frac{101}{294}$, $\frac{1}{294} a^{14} + \frac{1}{147} a^{12} - \frac{2}{147} a^{11} - \frac{10}{147} a^{10} + \frac{4}{147} a^{9} - \frac{1}{49} a^{8} + \frac{3}{49} a^{7} - \frac{2}{7} a^{6} - \frac{29}{147} a^{5} - \frac{61}{147} a^{4} + \frac{10}{147} a^{3} + \frac{10}{147} a^{2} - \frac{67}{147} a + \frac{65}{294}$, $\frac{1}{294} a^{15} - \frac{1}{147} a^{12} - \frac{1}{147} a^{11} - \frac{1}{147} a^{10} - \frac{2}{147} a^{9} - \frac{2}{49} a^{8} + \frac{1}{49} a^{7} + \frac{4}{21} a^{6} - \frac{55}{147} a^{5} + \frac{8}{147} a^{4} + \frac{71}{147} a^{3} - \frac{61}{147} a^{2} + \frac{109}{294} a - \frac{38}{147}$, $\frac{1}{294} a^{16} + \frac{1}{147} a^{12} - \frac{10}{147} a^{11} - \frac{1}{49} a^{10} + \frac{2}{147} a^{9} + \frac{1}{49} a^{8} - \frac{8}{147} a^{7} + \frac{2}{21} a^{6} - \frac{64}{147} a^{5} - \frac{32}{147} a^{4} + \frac{31}{147} a^{3} - \frac{5}{294} a^{2} + \frac{6}{49} a + \frac{50}{147}$, $\frac{1}{26754} a^{17} + \frac{1}{26754} a^{16} + \frac{1}{1029} a^{15} - \frac{5}{8918} a^{14} + \frac{2}{1911} a^{13} - \frac{4}{637} a^{12} - \frac{18}{637} a^{11} + \frac{671}{13377} a^{10} + \frac{135}{4459} a^{9} - \frac{2}{1029} a^{8} - \frac{58}{4459} a^{7} - \frac{108}{637} a^{6} - \frac{590}{1911} a^{5} - \frac{499}{1911} a^{4} + \frac{565}{8918} a^{3} - \frac{1675}{8918} a^{2} + \frac{6530}{13377} a + \frac{3061}{8918}$, $\frac{1}{26754} a^{18} + \frac{25}{26754} a^{16} - \frac{41}{26754} a^{15} + \frac{43}{26754} a^{14} - \frac{1}{1911} a^{13} - \frac{16}{1911} a^{12} + \frac{230}{13377} a^{11} - \frac{17}{637} a^{10} + \frac{99}{4459} a^{9} - \frac{148}{13377} a^{8} + \frac{121}{4459} a^{7} - \frac{188}{1911} a^{6} + \frac{19}{147} a^{5} + \frac{1741}{8918} a^{4} - \frac{103}{1911} a^{3} - \frac{3755}{26754} a^{2} - \frac{1717}{8918} a + \frac{3739}{26754}$, $\frac{1}{11932391016} a^{19} - \frac{204205}{11932391016} a^{18} + \frac{208571}{11932391016} a^{17} - \frac{13257427}{11932391016} a^{16} - \frac{5701901}{5966195508} a^{15} + \frac{3914941}{5966195508} a^{14} - \frac{490955}{568209096} a^{13} - \frac{20881541}{11932391016} a^{12} + \frac{168566407}{11932391016} a^{11} + \frac{90589585}{11932391016} a^{10} - \frac{65441759}{11932391016} a^{9} + \frac{102935113}{11932391016} a^{8} + \frac{436532053}{11932391016} a^{7} - \frac{205660859}{1704627288} a^{6} - \frac{2389882295}{5966195508} a^{5} + \frac{53227}{1565109} a^{4} + \frac{250332893}{1491548877} a^{3} - \frac{2491395253}{5966195508} a^{2} - \frac{4364726251}{11932391016} a + \frac{1133194535}{11932391016}$, $\frac{1}{40323182977578622327732134777383901504} a^{20} - \frac{175311528233701863122567813}{10080795744394655581933033694345975376} a^{19} + \frac{15410868194772255712003469988137}{960075785180443388755527018509140512} a^{18} + \frac{23187963253838056643872389402355}{3360265248131551860644344564781991792} a^{17} - \frac{13922433465539927998488738887713483}{13441060992526207442577378259127967168} a^{16} + \frac{177685817675698608092858425290121}{480037892590221694377763509254570256} a^{15} + \frac{208328766302786225355787901586217}{443111900852512333271781700850372544} a^{14} - \frac{3500528030282665826250024656164535}{10080795744394655581933033694345975376} a^{13} - \frac{521258980231513357766748418724323}{80970246942928960497454085898361248} a^{12} - \frac{477517474928074955539717779535460}{30002368286888855898610219328410641} a^{11} + \frac{232658028544212371008053183036360331}{6720530496263103721288689129563983584} a^{10} - \frac{730983396788463357091356519405295}{19146810530664113166064641394769184} a^{9} + \frac{34616871631178213800529713707278303}{960075785180443388755527018509140512} a^{8} - \frac{1377914514689547468038970793074716}{30002368286888855898610219328410641} a^{7} + \frac{18033808807637763575118289961180551237}{40323182977578622327732134777383901504} a^{6} - \frac{94610262865033243073418256528618679}{1550891652983793166451235952976303904} a^{5} + \frac{201134300733183572070849668810200223}{480037892590221694377763509254570256} a^{4} - \frac{919069870133207744125896981307390471}{2240176832087701240429563043187994528} a^{3} - \frac{841956440870549661826335330869864717}{3101783305967586332902471905952607808} a^{2} - \frac{13821116412198666587264614486577902}{30002368286888855898610219328410641} a - \frac{985807889584880796927980721991695215}{5760454711082660332533162111054843072}$

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:  \( 60294843557900000000000 \) (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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{7}$ $21$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R $21$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $21$ $21$ $21$

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
$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$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.7.0.1$x^{7} - 10 x + 2$$1$$7$$0$$C_7$$[\ ]^{7}$
13.7.0.1$x^{7} - 10 x + 2$$1$$7$$0$$C_7$$[\ ]^{7}$
305624493259Data not computed