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Properties

Label 20.0.53793575578...3125.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{16}\cdot 5^{12}\cdot 13^{15}$
Root discriminant $43.30$
Ramified primes $3, 5, 13$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5136, -10752, 15840, 942, -13197, -5128, 52537, 122101, 130902, 95285, 50642, 17107, 2799, -627, -769, -308, -29, -24, 14, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 14*x^18 - 24*x^17 - 29*x^16 - 308*x^15 - 769*x^14 - 627*x^13 + 2799*x^12 + 17107*x^11 + 50642*x^10 + 95285*x^9 + 130902*x^8 + 122101*x^7 + 52537*x^6 - 5128*x^5 - 13197*x^4 + 942*x^3 + 15840*x^2 - 10752*x + 5136)
 
gp: K = bnfinit(x^20 - x^19 + 14*x^18 - 24*x^17 - 29*x^16 - 308*x^15 - 769*x^14 - 627*x^13 + 2799*x^12 + 17107*x^11 + 50642*x^10 + 95285*x^9 + 130902*x^8 + 122101*x^7 + 52537*x^6 - 5128*x^5 - 13197*x^4 + 942*x^3 + 15840*x^2 - 10752*x + 5136, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 14 x^{18} - 24 x^{17} - 29 x^{16} - 308 x^{15} - 769 x^{14} - 627 x^{13} + 2799 x^{12} + 17107 x^{11} + 50642 x^{10} + 95285 x^{9} + 130902 x^{8} + 122101 x^{7} + 52537 x^{6} - 5128 x^{5} - 13197 x^{4} + 942 x^{3} + 15840 x^{2} - 10752 x + 5136 \)

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:  \(537935755789407686830516845703125=3^{16}\cdot 5^{12}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.30$
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:  $3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{15} - \frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{4} a^{10} - \frac{1}{24} a^{9} + \frac{5}{24} a^{7} - \frac{1}{2} a^{6} + \frac{11}{24} a^{5} + \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{16} - \frac{1}{12} a^{14} - \frac{1}{12} a^{12} - \frac{1}{4} a^{11} - \frac{1}{24} a^{10} + \frac{5}{24} a^{8} + \frac{11}{24} a^{6} + \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{24} a^{17} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{5}{24} a^{11} + \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{11}{24} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{18} - \frac{1}{4} a^{13} - \frac{5}{24} a^{12} + \frac{1}{8} a^{10} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{5}{24} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{31104147500913166859606667274366549604099448} a^{19} + \frac{6524521026124176290812422577109130547802}{3888018437614145857450833409295818700512431} a^{18} + \frac{60114011958981442423800446568720782961493}{7776036875228291714901666818591637401024862} a^{17} - \frac{281840049884181459954753224346280037615209}{15552073750456583429803333637183274802049724} a^{16} + \frac{11276006930860578067908337871099666732322}{1296006145871381952483611136431939566837477} a^{15} - \frac{9410904957154429631354046879539133133901}{7776036875228291714901666818591637401024862} a^{14} - \frac{4792387867542123020177053816276780155881573}{31104147500913166859606667274366549604099448} a^{13} + \frac{655465363397973269174272192057334877769378}{3888018437614145857450833409295818700512431} a^{12} - \frac{5427781229538664709805594191699766124508321}{31104147500913166859606667274366549604099448} a^{11} - \frac{249429817530816031300562271633568226644445}{15552073750456583429803333637183274802049724} a^{10} + \frac{770978023104900081776353757611939410885433}{10368049166971055619868889091455516534699816} a^{9} + \frac{1830359033656622461104930954285882230587657}{15552073750456583429803333637183274802049724} a^{8} + \frac{6950778482183352039868812089749774929321811}{31104147500913166859606667274366549604099448} a^{7} + \frac{737051439476501441702537945242315343970375}{5184024583485527809934444545727758267349908} a^{6} - \frac{1535004296333860204725390995973632542000663}{3888018437614145857450833409295818700512431} a^{5} + \frac{337994361260255548828898777651683823213204}{1296006145871381952483611136431939566837477} a^{4} - \frac{1359953504555432998778215779324923522808901}{5184024583485527809934444545727758267349908} a^{3} - \frac{1729176023476930247567026734777702408653405}{5184024583485527809934444545727758267349908} a^{2} + \frac{845807277419424000045180070598806767366473}{2592012291742763904967222272863879133674954} a - \frac{156106632175114847036308666315593899336441}{1296006145871381952483611136431939566837477}$

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:  \( 315151166.6618722 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

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 $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 5.1.1711125.1, 10.2.38063333953125.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ $20$

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
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
5Data not computed
13Data not computed