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Properties

Label 20.0.17010572286...7957.1
Degree $20$
Signature $[0, 10]$
Discriminant $7^{16}\cdot 13^{15}$
Root discriminant $32.47$
Ramified primes $7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![243, 1215, 2700, 4455, 7581, 10639, 8307, 2598, -1200, -4550, -2242, 1599, -62, 206, 260, -345, 210, -90, 29, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 29*x^18 - 90*x^17 + 210*x^16 - 345*x^15 + 260*x^14 + 206*x^13 - 62*x^12 + 1599*x^11 - 2242*x^10 - 4550*x^9 - 1200*x^8 + 2598*x^7 + 8307*x^6 + 10639*x^5 + 7581*x^4 + 4455*x^3 + 2700*x^2 + 1215*x + 243)
 
gp: K = bnfinit(x^20 - 7*x^19 + 29*x^18 - 90*x^17 + 210*x^16 - 345*x^15 + 260*x^14 + 206*x^13 - 62*x^12 + 1599*x^11 - 2242*x^10 - 4550*x^9 - 1200*x^8 + 2598*x^7 + 8307*x^6 + 10639*x^5 + 7581*x^4 + 4455*x^3 + 2700*x^2 + 1215*x + 243, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 29 x^{18} - 90 x^{17} + 210 x^{16} - 345 x^{15} + 260 x^{14} + 206 x^{13} - 62 x^{12} + 1599 x^{11} - 2242 x^{10} - 4550 x^{9} - 1200 x^{8} + 2598 x^{7} + 8307 x^{6} + 10639 x^{5} + 7581 x^{4} + 4455 x^{3} + 2700 x^{2} + 1215 x + 243 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1701057228680302987747119277957=7^{16}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.47$
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 Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{26} a^{12} - \frac{3}{26} a^{11} + \frac{1}{26} a^{10} - \frac{2}{13} a^{9} - \frac{6}{13} a^{8} - \frac{11}{26} a^{7} + \frac{11}{26} a^{6} + \frac{5}{26} a^{5} - \frac{5}{13} a^{4} - \frac{2}{13} a^{3} + \frac{9}{26} a^{2} - \frac{9}{26} a + \frac{1}{26}$, $\frac{1}{26} a^{13} + \frac{5}{26} a^{11} - \frac{1}{26} a^{10} + \frac{1}{13} a^{9} + \frac{5}{26} a^{8} + \frac{2}{13} a^{7} - \frac{1}{26} a^{6} + \frac{5}{26} a^{5} - \frac{4}{13} a^{4} - \frac{3}{26} a^{3} - \frac{4}{13} a^{2} - \frac{1}{2} a + \frac{3}{26}$, $\frac{1}{26} a^{14} + \frac{1}{26} a^{11} - \frac{3}{26} a^{10} - \frac{1}{26} a^{9} + \frac{6}{13} a^{8} + \frac{1}{13} a^{7} - \frac{11}{26} a^{6} - \frac{7}{26} a^{5} - \frac{5}{26} a^{4} + \frac{6}{13} a^{3} - \frac{3}{13} a^{2} + \frac{9}{26} a - \frac{5}{26}$, $\frac{1}{26} a^{15} - \frac{1}{13} a^{10} - \frac{5}{13} a^{9} - \frac{6}{13} a^{8} + \frac{4}{13} a^{6} - \frac{5}{13} a^{5} - \frac{2}{13} a^{4} - \frac{1}{13} a^{3} + \frac{2}{13} a - \frac{1}{26}$, $\frac{1}{156} a^{16} - \frac{1}{156} a^{15} - \frac{1}{156} a^{14} - \frac{1}{52} a^{13} + \frac{7}{52} a^{11} - \frac{7}{39} a^{10} + \frac{71}{156} a^{9} + \frac{37}{156} a^{8} + \frac{6}{13} a^{7} + \frac{35}{156} a^{6} - \frac{1}{78} a^{5} + \frac{19}{52} a^{4} + \frac{17}{52} a^{3} + \frac{5}{13} a^{2} + \frac{16}{39} a + \frac{25}{52}$, $\frac{1}{468} a^{17} - \frac{1}{468} a^{16} + \frac{5}{468} a^{15} + \frac{1}{156} a^{14} - \frac{1}{78} a^{13} - \frac{1}{156} a^{12} + \frac{49}{234} a^{11} + \frac{101}{468} a^{10} + \frac{211}{468} a^{9} + \frac{29}{78} a^{8} - \frac{181}{468} a^{7} - \frac{50}{117} a^{6} + \frac{1}{12} a^{5} - \frac{61}{156} a^{4} - \frac{1}{2} a^{3} + \frac{43}{117} a^{2} - \frac{11}{52} a$, $\frac{1}{37064196} a^{18} + \frac{5741}{9266049} a^{17} - \frac{3742}{9266049} a^{16} - \frac{5377}{3088683} a^{15} + \frac{228499}{12354732} a^{14} - \frac{212449}{12354732} a^{13} - \frac{669727}{37064196} a^{12} + \frac{4586405}{37064196} a^{11} - \frac{4374271}{18532098} a^{10} - \frac{371897}{1372748} a^{9} + \frac{2543795}{37064196} a^{8} + \frac{8349229}{37064196} a^{7} + \frac{1107701}{4118244} a^{6} - \frac{2826581}{6177366} a^{5} + \frac{2042479}{4118244} a^{4} - \frac{1483886}{9266049} a^{3} - \frac{2159347}{12354732} a^{2} - \frac{613745}{4118244} a - \frac{30769}{686374}$, $\frac{1}{617647052103520545756} a^{19} - \frac{4836004215409}{617647052103520545756} a^{18} - \frac{635867283892263223}{617647052103520545756} a^{17} + \frac{118088221698862273}{51470587675293378813} a^{16} - \frac{22824398412113161}{1821967705320119604} a^{15} + \frac{582982133517569063}{51470587675293378813} a^{14} - \frac{2196893880854870221}{617647052103520545756} a^{13} - \frac{10782171892155479353}{617647052103520545756} a^{12} + \frac{47166391942663817279}{308823526051760272878} a^{11} + \frac{13175569505888707829}{102941175350586757626} a^{10} + \frac{121149668431469297029}{308823526051760272878} a^{9} - \frac{3842876363741044181}{14363884932640012692} a^{8} + \frac{52626020627229375043}{205882350701173515252} a^{7} + \frac{28139466835686393601}{102941175350586757626} a^{6} - \frac{2782033733720650255}{5718954186143708757} a^{5} + \frac{12273971471437312981}{617647052103520545756} a^{4} + \frac{738110314110393317}{17156862558431126271} a^{3} - \frac{1184556659869230527}{2639517316681711734} a^{2} + \frac{1491923351219793571}{11437908372287417514} a - \frac{75352705535786445}{586559403707047052}$

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:  $9$
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:  \( 9407573.54607 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 5.1.5274997.1 x5, 10.2.361732713550117.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.5274997.1
Degree 10 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
7Data not computed
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$