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Properties

Label 21.5.37877966044...6336.2
Degree $21$
Signature $[5, 8]$
Discriminant $2^{24}\cdot 11^{18}\cdot 67^{8}$
Root discriminant $85.57$
Ramified primes $2, 11, 67$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $\PSL(3,4)$ (as 21T67)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9397944, 29752822, -9878676, -35335382, 21732094, 22487749, -21914184, -4343266, 11070150, -2612752, -2046920, 1174704, 42258, -181415, 35134, 10924, -4806, 49, 250, -34, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 - 34*x^19 + 250*x^18 + 49*x^17 - 4806*x^16 + 10924*x^15 + 35134*x^14 - 181415*x^13 + 42258*x^12 + 1174704*x^11 - 2046920*x^10 - 2612752*x^9 + 11070150*x^8 - 4343266*x^7 - 21914184*x^6 + 22487749*x^5 + 21732094*x^4 - 35335382*x^3 - 9878676*x^2 + 29752822*x - 9397944)
 
gp: K = bnfinit(x^21 - 4*x^20 - 34*x^19 + 250*x^18 + 49*x^17 - 4806*x^16 + 10924*x^15 + 35134*x^14 - 181415*x^13 + 42258*x^12 + 1174704*x^11 - 2046920*x^10 - 2612752*x^9 + 11070150*x^8 - 4343266*x^7 - 21914184*x^6 + 22487749*x^5 + 21732094*x^4 - 35335382*x^3 - 9878676*x^2 + 29752822*x - 9397944, 1)
 

Normalized defining polynomial

\( x^{21} - 4 x^{20} - 34 x^{19} + 250 x^{18} + 49 x^{17} - 4806 x^{16} + 10924 x^{15} + 35134 x^{14} - 181415 x^{13} + 42258 x^{12} + 1174704 x^{11} - 2046920 x^{10} - 2612752 x^{9} + 11070150 x^{8} - 4343266 x^{7} - 21914184 x^{6} + 22487749 x^{5} + 21732094 x^{4} - 35335382 x^{3} - 9878676 x^{2} + 29752822 x - 9397944 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(37877966044500313823897559260772576526336=2^{24}\cdot 11^{18}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $85.57$
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:  $2, 11, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{16} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{59711674143952451480517531386424925636072052562398016006395320503077816} a^{20} + \frac{3056115344834877358044346045703769914062867151425762528115517506764971}{59711674143952451480517531386424925636072052562398016006395320503077816} a^{19} - \frac{7863374819359903637689297584978551612237824970463610112643439621202845}{59711674143952451480517531386424925636072052562398016006395320503077816} a^{18} - \frac{381912630464892649302316370548826519431931013345502809724055296782025}{59711674143952451480517531386424925636072052562398016006395320503077816} a^{17} - \frac{9144936170466518357829554964295352110254130268455130323062473592658619}{29855837071976225740258765693212462818036026281199008003197660251538908} a^{16} - \frac{1761452651106464957513439317305197307427768246987526789972817053204929}{7463959267994056435064691423303115704509006570299752000799415062884727} a^{15} - \frac{2799282135625965048647654484608716697746430639573767539611364657637535}{14927918535988112870129382846606231409018013140599504001598830125769454} a^{14} + \frac{6622564963060971627865067836562188152992546025100641365871001622242849}{29855837071976225740258765693212462818036026281199008003197660251538908} a^{13} + \frac{5098015031070267145626318098643959392744324511443872963517688933314551}{59711674143952451480517531386424925636072052562398016006395320503077816} a^{12} - \frac{6908823237725437663383897298373891409626330450092436096273002843462109}{59711674143952451480517531386424925636072052562398016006395320503077816} a^{11} - \frac{27636142992680758517140691156782387740556302211354162664235283424745195}{59711674143952451480517531386424925636072052562398016006395320503077816} a^{10} - \frac{22962757673062477373541692229174763224281350986141077722679923933822245}{59711674143952451480517531386424925636072052562398016006395320503077816} a^{9} + \frac{20650866544027858219958748920044454003583646418217305635816797011399877}{59711674143952451480517531386424925636072052562398016006395320503077816} a^{8} - \frac{23461018299001092368386715510357808739668041656125533434547183634833599}{59711674143952451480517531386424925636072052562398016006395320503077816} a^{7} + \frac{24811223681760638007844421023642435754222803985090283038648565601841853}{59711674143952451480517531386424925636072052562398016006395320503077816} a^{6} + \frac{3408209155836209406690699653721524751750319194442107181121721224943339}{59711674143952451480517531386424925636072052562398016006395320503077816} a^{5} + \frac{5445667338423422992675323535065669187039009046960120772593309130165957}{29855837071976225740258765693212462818036026281199008003197660251538908} a^{4} - \frac{4634702316243718846617186440008737762013987789137408634212710173349561}{14927918535988112870129382846606231409018013140599504001598830125769454} a^{3} - \frac{13525628125774271507699552741564332611374051789491914936021887743582153}{29855837071976225740258765693212462818036026281199008003197660251538908} a^{2} + \frac{625309663089738604969281823040114976119590907272909632695725369555495}{29855837071976225740258765693212462818036026281199008003197660251538908} a + \frac{3376597462654690936647894366665694410878771910942596229251778872193537}{7463959267994056435064691423303115704509006570299752000799415062884727}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$ (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:  $12$
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:  \( 10194481907400 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\PSL(3,4)$ (as 21T67):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 20160
The 10 conjugacy class representatives for $\PSL(3,4)$
Character table for $\PSL(3,4)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Arithmetically equvalently 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.12.18.59$x^{12} - 2 x^{11} + 6 x^{10} + 4 x^{9} + 6 x^{8} + 12 x^{7} - 4 x^{6} - 8 x^{3} + 16 x^{2} - 8$$4$$3$$18$$A_4$$[2, 2]^{3}$
$11$11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.8.4.1$x^{8} + 17956 x^{4} - 300763 x^{2} + 80604484$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
67.8.4.1$x^{8} + 17956 x^{4} - 300763 x^{2} + 80604484$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$