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Properties

Label 20.0.26062314028...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{10}\cdot 5^{22}\cdot 7^{10}$
Root discriminant $46.86$
Ramified primes $2, 3, 5, 7$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6304, 19840, 19840, 38120, 29440, -13974, 31860, -70620, 29600, -49850, 25755, -15900, 13495, -2880, 3650, -332, 500, -20, 35, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 35*x^18 - 20*x^17 + 500*x^16 - 332*x^15 + 3650*x^14 - 2880*x^13 + 13495*x^12 - 15900*x^11 + 25755*x^10 - 49850*x^9 + 29600*x^8 - 70620*x^7 + 31860*x^6 - 13974*x^5 + 29440*x^4 + 38120*x^3 + 19840*x^2 + 19840*x + 6304)
 
gp: K = bnfinit(x^20 + 35*x^18 - 20*x^17 + 500*x^16 - 332*x^15 + 3650*x^14 - 2880*x^13 + 13495*x^12 - 15900*x^11 + 25755*x^10 - 49850*x^9 + 29600*x^8 - 70620*x^7 + 31860*x^6 - 13974*x^5 + 29440*x^4 + 38120*x^3 + 19840*x^2 + 19840*x + 6304, 1)
 

Normalized defining polynomial

\( x^{20} + 35 x^{18} - 20 x^{17} + 500 x^{16} - 332 x^{15} + 3650 x^{14} - 2880 x^{13} + 13495 x^{12} - 15900 x^{11} + 25755 x^{10} - 49850 x^{9} + 29600 x^{8} - 70620 x^{7} + 31860 x^{6} - 13974 x^{5} + 29440 x^{4} + 38120 x^{3} + 19840 x^{2} + 19840 x + 6304 \)

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:  \(2606231402843906250000000000000000=2^{16}\cdot 3^{10}\cdot 5^{22}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.86$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{16} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{8} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{17} - \frac{1}{4} a^{11} - \frac{3}{8} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4}$, $\frac{1}{150704} a^{18} + \frac{255}{150704} a^{17} - \frac{482}{9419} a^{16} + \frac{2495}{37676} a^{15} + \frac{4269}{37676} a^{14} + \frac{4625}{18838} a^{13} - \frac{2595}{75352} a^{12} + \frac{9511}{75352} a^{11} - \frac{11319}{150704} a^{10} + \frac{59091}{150704} a^{9} - \frac{4559}{37676} a^{8} - \frac{24099}{75352} a^{7} + \frac{14965}{75352} a^{6} + \frac{3145}{75352} a^{5} + \frac{18509}{75352} a^{4} - \frac{2576}{9419} a^{3} - \frac{2913}{9419} a^{2} + \frac{2693}{9419} a + \frac{1190}{9419}$, $\frac{1}{98996337755741658312556418537688986672} a^{19} + \frac{112024175202509333691102049184637}{49498168877870829156278209268844493336} a^{18} + \frac{4583049703329118722672910683882392787}{98996337755741658312556418537688986672} a^{17} + \frac{1515454396103018923957715199433274049}{24749084438935414578139104634422246668} a^{16} - \frac{413678728073507788248347327425216755}{24749084438935414578139104634422246668} a^{15} - \frac{1543270503973985120617393788569431789}{12374542219467707289069552317211123334} a^{14} + \frac{11314550983624714726195800837647894383}{49498168877870829156278209268844493336} a^{13} - \frac{357680212766019254893082631592080941}{6187271109733853644534776158605561667} a^{12} + \frac{3663754807166450736516509064341672435}{98996337755741658312556418537688986672} a^{11} - \frac{10818350504981431296839217427050921471}{49498168877870829156278209268844493336} a^{10} - \frac{3001679592128851643788341011432890553}{98996337755741658312556418537688986672} a^{9} + \frac{20107564078315862364474781699630703017}{49498168877870829156278209268844493336} a^{8} - \frac{1728115675661582445072301615981341907}{6187271109733853644534776158605561667} a^{7} - \frac{2847046019716719273046200612993582321}{24749084438935414578139104634422246668} a^{6} - \frac{467531765344851645767687387185683007}{951887863035977483774580947477778718} a^{5} + \frac{10174725313959918424032731183309858445}{49498168877870829156278209268844493336} a^{4} + \frac{3023825224128463133376543594503360277}{12374542219467707289069552317211123334} a^{3} + \frac{14412587767066261284376481119751085}{475943931517988741887290473738889359} a^{2} - \frac{2422108161241480675309829099132516086}{6187271109733853644534776158605561667} a - \frac{2659370655586950242962283031243391779}{6187271109733853644534776158605561667}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1949607169.9383223 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-7}, \sqrt{-15})\), 5.1.50000.1, 10.0.3037500000000.2, 10.0.42017500000000.4, 10.2.51051262500000000.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 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 R R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$