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Properties

Label 20.4.17095188714...4121.1
Degree $20$
Signature $[4, 8]$
Discriminant $13^{2}\cdot 97^{2}\cdot 401^{10}$
Root discriminant $40.89$
Ramified primes $13, 97, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T347

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![289, -1564, -7139, -19059, -53001, -78624, -65888, -28354, -16011, -4394, -4963, 2768, -1251, 443, 365, -211, 173, -51, 22, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 22*x^18 - 51*x^17 + 173*x^16 - 211*x^15 + 365*x^14 + 443*x^13 - 1251*x^12 + 2768*x^11 - 4963*x^10 - 4394*x^9 - 16011*x^8 - 28354*x^7 - 65888*x^6 - 78624*x^5 - 53001*x^4 - 19059*x^3 - 7139*x^2 - 1564*x + 289)
 
gp: K = bnfinit(x^20 - 4*x^19 + 22*x^18 - 51*x^17 + 173*x^16 - 211*x^15 + 365*x^14 + 443*x^13 - 1251*x^12 + 2768*x^11 - 4963*x^10 - 4394*x^9 - 16011*x^8 - 28354*x^7 - 65888*x^6 - 78624*x^5 - 53001*x^4 - 19059*x^3 - 7139*x^2 - 1564*x + 289, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 22 x^{18} - 51 x^{17} + 173 x^{16} - 211 x^{15} + 365 x^{14} + 443 x^{13} - 1251 x^{12} + 2768 x^{11} - 4963 x^{10} - 4394 x^{9} - 16011 x^{8} - 28354 x^{7} - 65888 x^{6} - 78624 x^{5} - 53001 x^{4} - 19059 x^{3} - 7139 x^{2} - 1564 x + 289 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(170951887142655083512160513274121=13^{2}\cdot 97^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.89$
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:  $13, 97, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{747} a^{18} + \frac{22}{747} a^{17} + \frac{2}{747} a^{16} - \frac{36}{83} a^{15} + \frac{92}{249} a^{14} + \frac{71}{747} a^{13} + \frac{19}{83} a^{12} + \frac{23}{83} a^{11} + \frac{1}{83} a^{10} - \frac{22}{747} a^{9} + \frac{38}{83} a^{8} + \frac{341}{747} a^{7} + \frac{298}{747} a^{6} + \frac{128}{747} a^{5} + \frac{64}{747} a^{4} - \frac{116}{249} a^{3} + \frac{161}{747} a^{2} - \frac{89}{747} a - \frac{185}{747}$, $\frac{1}{84843174588866491031891782560743614262714349} a^{19} + \frac{6055230163772498300736344867622794893594}{28281058196288830343963927520247871420904783} a^{18} + \frac{8700705525955534373873498588633866351161058}{84843174588866491031891782560743614262714349} a^{17} - \frac{1769804538149746486352086787659003192599730}{4990774975815675943052457797690800838983197} a^{16} + \frac{1807845464225237325191853160658127180085790}{28281058196288830343963927520247871420904783} a^{15} + \frac{40151440326306395243244800209663884373798673}{84843174588866491031891782560743614262714349} a^{14} - \frac{15873851021644569361178261866385813999455034}{84843174588866491031891782560743614262714349} a^{13} - \frac{3379589680491145736944365732405806228506751}{9427019398762943447987975840082623806968261} a^{12} + \frac{641729036155658793767350362126840888647443}{3142339799587647815995991946694207935656087} a^{11} - \frac{633925030691957782968495781690959069348343}{2293058772672067325186264393533611196289577} a^{10} - \frac{36709914740879343725268385680796959228481520}{84843174588866491031891782560743614262714349} a^{9} + \frac{8927199006152565745951293424994400061795508}{84843174588866491031891782560743614262714349} a^{8} - \frac{19646170663902281379635112116518239855044809}{84843174588866491031891782560743614262714349} a^{7} - \frac{18652551807649984562742719439483609032320253}{84843174588866491031891782560743614262714349} a^{6} - \frac{37452036826048693185622488927302690459907385}{84843174588866491031891782560743614262714349} a^{5} + \frac{35680448949455368584310384400000988313007318}{84843174588866491031891782560743614262714349} a^{4} - \frac{6462501262608605588280481751056968936318130}{84843174588866491031891782560743614262714349} a^{3} + \frac{8630364841384186043732926302722479300541969}{84843174588866491031891782560743614262714349} a^{2} + \frac{12177522373225918779451589554602403231213555}{28281058196288830343963927520247871420904783} a + \frac{380909325829841788792716288962591430107884}{4990774975815675943052457797690800838983197}$

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:  \( 120168173.631 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T347:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n347 are not computed
Character table for t20n347 is not computed

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
401Data not computed