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Properties

Label 21.21.1966542379...1248.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 7^{29}\cdot 13^{12}$
Root discriminant $115.24$
Ramified primes $2, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_7:(C_3\times D_7)$ (as 21T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-43, -2184, 17031, 284319, -2593367, 3446688, 9478959, -15807117, -11164454, 16233798, 8210370, -4812654, -2160550, 701288, 268223, -59164, -17227, 2989, 553, -84, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 84*x^19 + 553*x^18 + 2989*x^17 - 17227*x^16 - 59164*x^15 + 268223*x^14 + 701288*x^13 - 2160550*x^12 - 4812654*x^11 + 8210370*x^10 + 16233798*x^9 - 11164454*x^8 - 15807117*x^7 + 9478959*x^6 + 3446688*x^5 - 2593367*x^4 + 284319*x^3 + 17031*x^2 - 2184*x - 43)
 
gp: K = bnfinit(x^21 - 7*x^20 - 84*x^19 + 553*x^18 + 2989*x^17 - 17227*x^16 - 59164*x^15 + 268223*x^14 + 701288*x^13 - 2160550*x^12 - 4812654*x^11 + 8210370*x^10 + 16233798*x^9 - 11164454*x^8 - 15807117*x^7 + 9478959*x^6 + 3446688*x^5 - 2593367*x^4 + 284319*x^3 + 17031*x^2 - 2184*x - 43, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 84 x^{19} + 553 x^{18} + 2989 x^{17} - 17227 x^{16} - 59164 x^{15} + 268223 x^{14} + 701288 x^{13} - 2160550 x^{12} - 4812654 x^{11} + 8210370 x^{10} + 16233798 x^{9} - 11164454 x^{8} - 15807117 x^{7} + 9478959 x^{6} + 3446688 x^{5} - 2593367 x^{4} + 284319 x^{3} + 17031 x^{2} - 2184 x - 43 \)

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:  \(19665423796876602841207515435602265497141248=2^{18}\cdot 7^{29}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $115.24$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2}$, $\frac{1}{10} a^{15} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{7} - \frac{2}{5} a^{5} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{3}{10} a + \frac{1}{5}$, $\frac{1}{10} a^{16} + \frac{1}{5} a^{14} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{8} - \frac{2}{5} a^{6} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{3}{10} a^{2} + \frac{1}{5} a$, $\frac{1}{20} a^{17} - \frac{1}{20} a^{16} - \frac{1}{4} a^{14} - \frac{1}{5} a^{13} + \frac{3}{10} a^{12} - \frac{3}{10} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{3}{10} a^{6} - \frac{3}{10} a^{5} - \frac{1}{2} a^{4} - \frac{1}{20} a^{3} + \frac{3}{20} a^{2} - \frac{3}{10} a + \frac{1}{20}$, $\frac{1}{1300} a^{18} + \frac{6}{325} a^{17} - \frac{59}{1300} a^{16} - \frac{51}{1300} a^{15} - \frac{127}{1300} a^{14} + \frac{113}{650} a^{13} + \frac{31}{325} a^{12} - \frac{107}{650} a^{11} - \frac{124}{325} a^{10} + \frac{121}{325} a^{9} - \frac{28}{65} a^{8} + \frac{269}{650} a^{7} - \frac{19}{65} a^{6} + \frac{96}{325} a^{5} + \frac{77}{260} a^{4} + \frac{109}{650} a^{3} - \frac{113}{260} a^{2} + \frac{361}{1300} a - \frac{97}{1300}$, $\frac{1}{5200} a^{19} + \frac{1}{5200} a^{18} + \frac{3}{400} a^{17} - \frac{31}{1300} a^{16} - \frac{127}{2600} a^{15} + \frac{937}{5200} a^{14} + \frac{1233}{2600} a^{13} + \frac{937}{2600} a^{12} + \frac{783}{2600} a^{11} + \frac{12}{325} a^{10} + \frac{327}{1300} a^{9} - \frac{831}{2600} a^{8} + \frac{773}{2600} a^{7} + \frac{34}{325} a^{6} + \frac{1953}{5200} a^{5} + \frac{983}{5200} a^{4} + \frac{2091}{5200} a^{3} - \frac{277}{2600} a^{2} - \frac{27}{65} a - \frac{2319}{5200}$, $\frac{1}{418187861490821071749988109000372079416850935856400} a^{20} + \frac{3270958092945929587577964825236985398576269361}{418187861490821071749988109000372079416850935856400} a^{19} + \frac{892661171395590746256507896647398987392197911}{83637572298164214349997621800074415883370187171280} a^{18} - \frac{1719032729445599275961140558850451120733722549}{160841485188777335288456965000143107468019590714} a^{17} - \frac{5524543522591702628649928693565292523373185308959}{209093930745410535874994054500186039708425467928200} a^{16} + \frac{9153417697945523496511929106097500276579754265861}{418187861490821071749988109000372079416850935856400} a^{15} + \frac{44237127635794887846961681898622899072599098731957}{209093930745410535874994054500186039708425467928200} a^{14} + \frac{12000609485307920204486083979402515707228960194401}{41818786149082107174998810900037207941685093585640} a^{13} - \frac{7871988379285116146492053071573881100730392825513}{41818786149082107174998810900037207941685093585640} a^{12} - \frac{6331053987204516017716851701337036078235578657137}{26136741343176316984374256812523254963553183491025} a^{11} - \frac{9041930609230593426131637867667694284463994853017}{104546965372705267937497027250093019854212733964100} a^{10} + \frac{40864533076590567533169112884117058125204239376481}{209093930745410535874994054500186039708425467928200} a^{9} - \frac{53193521807214365350368710591804043882517797930427}{209093930745410535874994054500186039708425467928200} a^{8} - \frac{1541148752144256378041656744897448921745199404011}{52273482686352633968748513625046509927106366982050} a^{7} + \frac{167128883992618140393787111109683654342897912360393}{418187861490821071749988109000372079416850935856400} a^{6} + \frac{93992993681317898952032329159795037861749775087247}{418187861490821071749988109000372079416850935856400} a^{5} + \frac{12537179156896826576038446850356175660905327512427}{32168297037755467057691393000028621493603918142800} a^{4} - \frac{32766771848215088370628929020316405134836273510043}{209093930745410535874994054500186039708425467928200} a^{3} - \frac{3958778141260944709539107176328407889325103279889}{10454696537270526793749702725009301985421273396410} a^{2} - \frac{33500805743526507310138194903150934506360910937563}{418187861490821071749988109000372079416850935856400} a - \frac{25588693448030709641581909055319805963107543012149}{52273482686352633968748513625046509927106366982050}$

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:  \( 1720832574690000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7:(C_3\times D_7)$ (as 21T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 294
The 22 conjugacy class representatives for $C_7:(C_3\times D_7)$
Character table for $C_7:(C_3\times D_7)$ 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 21 sibling: 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 $21$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ $21$ $21$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ $21$ $21$ $21$

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$$[\ ]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
7Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$