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Properties

Label 21.3.26160559119...0703.2
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 13^{18}$
Root discriminant $43.54$
Ramified primes $7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_7$ (as 21T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19264, -133952, 70896, 215936, -198072, -83972, 150977, -45562, -15492, 18797, -6256, -1701, 988, 847, -718, 201, -32, 29, -34, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 - 32*x^16 + 201*x^15 - 718*x^14 + 847*x^13 + 988*x^12 - 1701*x^11 - 6256*x^10 + 18797*x^9 - 15492*x^8 - 45562*x^7 + 150977*x^6 - 83972*x^5 - 198072*x^4 + 215936*x^3 + 70896*x^2 - 133952*x + 19264)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 - 32*x^16 + 201*x^15 - 718*x^14 + 847*x^13 + 988*x^12 - 1701*x^11 - 6256*x^10 + 18797*x^9 - 15492*x^8 - 45562*x^7 + 150977*x^6 - 83972*x^5 - 198072*x^4 + 215936*x^3 + 70896*x^2 - 133952*x + 19264, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} - 32 x^{16} + 201 x^{15} - 718 x^{14} + 847 x^{13} + 988 x^{12} - 1701 x^{11} - 6256 x^{10} + 18797 x^{9} - 15492 x^{8} - 45562 x^{7} + 150977 x^{6} - 83972 x^{5} - 198072 x^{4} + 215936 x^{3} + 70896 x^{2} - 133952 x + 19264 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-26160559119874379648562947375700703=-\,7^{17}\cdot 13^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.54$
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:  $7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{172} a^{16} + \frac{19}{172} a^{15} - \frac{5}{86} a^{14} + \frac{15}{172} a^{13} - \frac{3}{43} a^{12} - \frac{11}{172} a^{11} - \frac{7}{43} a^{10} - \frac{13}{172} a^{9} + \frac{8}{43} a^{8} - \frac{3}{172} a^{7} + \frac{14}{43} a^{6} + \frac{11}{172} a^{5} + \frac{33}{86} a^{4} - \frac{13}{172} a^{3} + \frac{15}{172} a^{2} + \frac{18}{43} a$, $\frac{1}{172} a^{17} + \frac{4}{43} a^{15} - \frac{5}{86} a^{14} + \frac{1}{43} a^{13} + \frac{1}{86} a^{12} - \frac{17}{86} a^{11} - \frac{10}{43} a^{10} - \frac{11}{86} a^{9} + \frac{17}{86} a^{8} - \frac{4}{43} a^{7} + \frac{11}{86} a^{6} + \frac{18}{43} a^{5} - \frac{5}{43} a^{4} - \frac{39}{172} a^{3} - \frac{21}{43} a^{2} + \frac{2}{43} a$, $\frac{1}{62608} a^{18} - \frac{127}{62608} a^{17} - \frac{95}{62608} a^{16} + \frac{569}{31304} a^{15} - \frac{6431}{62608} a^{14} + \frac{426}{3913} a^{13} + \frac{2549}{62608} a^{12} + \frac{875}{4472} a^{11} + \frac{1357}{8944} a^{10} + \frac{627}{15652} a^{9} - \frac{901}{4816} a^{8} - \frac{825}{7826} a^{7} - \frac{2703}{62608} a^{6} + \frac{807}{2236} a^{5} + \frac{887}{4472} a^{4} + \frac{2935}{8944} a^{3} - \frac{165}{2236} a^{2} - \frac{205}{2236} a - \frac{5}{26}$, $\frac{1}{125216} a^{19} - \frac{1}{125216} a^{18} - \frac{81}{125216} a^{17} + \frac{11}{15652} a^{16} + \frac{5917}{125216} a^{15} - \frac{99}{1456} a^{14} - \frac{6411}{125216} a^{13} - \frac{11}{86} a^{12} - \frac{2783}{17888} a^{11} + \frac{1275}{62608} a^{10} + \frac{677}{2912} a^{9} - \frac{12491}{62608} a^{8} - \frac{27679}{125216} a^{7} - \frac{3057}{8944} a^{6} + \frac{2283}{8944} a^{5} - \frac{2549}{17888} a^{4} - \frac{2105}{8944} a^{3} - \frac{45}{344} a^{2} - \frac{3}{43} a - \frac{3}{26}$, $\frac{1}{207887600403310915083504121019002688} a^{20} + \frac{392226377616737257057298949711}{207887600403310915083504121019002688} a^{19} + \frac{632784562454770997269422143127}{207887600403310915083504121019002688} a^{18} + \frac{15952935065432715655080603828417}{25985950050413864385438015127375336} a^{17} - \frac{332137251071673234955476342776003}{207887600403310915083504121019002688} a^{16} + \frac{3044875649005602203659034727579795}{103943800201655457541752060509501344} a^{15} - \frac{24428493743126974352697382943679659}{207887600403310915083504121019002688} a^{14} - \frac{84082387668142479495659896576209}{25985950050413864385438015127375336} a^{13} + \frac{32366112435113442001476838909474295}{207887600403310915083504121019002688} a^{12} + \frac{12251982317273470162946899269101583}{103943800201655457541752060509501344} a^{11} - \frac{23257761248461243945486238587440217}{207887600403310915083504121019002688} a^{10} + \frac{15067422180511387988774606712349977}{103943800201655457541752060509501344} a^{9} + \frac{27565363643128725524822981796298497}{207887600403310915083504121019002688} a^{8} - \frac{22532802815756973078037141340242083}{103943800201655457541752060509501344} a^{7} - \frac{42706811753240615636602763095601379}{103943800201655457541752060509501344} a^{6} + \frac{1393114630550789298247373731340867}{29698228629044416440500588717000384} a^{5} + \frac{419712675581573610631503635170123}{14849114314522208220250294358500192} a^{4} - \frac{1952005121588922363008244429807531}{7424557157261104110125147179250096} a^{3} + \frac{314875157617316861394023340763673}{928069644657638013765643397406262} a^{2} + \frac{865297688965495029352818652672789}{1856139289315276027531286794812524} a - \frac{2035339649120000948703490343148}{10791507496019046671693527876817}$

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:  $11$
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:  \( 3123124290.4668374 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_7$ (as 21T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.1.81124178863.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 7 sibling: data not computed
Degree 14 sibling: 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{21}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$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$13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$