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Properties

Label 20.12.2106183370...4848.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{10}\cdot 19^{12}\cdot 43^{11}$
Root discriminant $65.49$
Ramified primes $2, 19, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T423

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7775, 508220, 346947, -756560, -1691906, -159284, 2268740, 576352, -1454928, -267232, 512568, 38920, -104402, 4334, 11793, -2030, -574, 222, -5, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 5*x^18 + 222*x^17 - 574*x^16 - 2030*x^15 + 11793*x^14 + 4334*x^13 - 104402*x^12 + 38920*x^11 + 512568*x^10 - 267232*x^9 - 1454928*x^8 + 576352*x^7 + 2268740*x^6 - 159284*x^5 - 1691906*x^4 - 756560*x^3 + 346947*x^2 + 508220*x + 7775)
 
gp: K = bnfinit(x^20 - 8*x^19 - 5*x^18 + 222*x^17 - 574*x^16 - 2030*x^15 + 11793*x^14 + 4334*x^13 - 104402*x^12 + 38920*x^11 + 512568*x^10 - 267232*x^9 - 1454928*x^8 + 576352*x^7 + 2268740*x^6 - 159284*x^5 - 1691906*x^4 - 756560*x^3 + 346947*x^2 + 508220*x + 7775, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} - 5 x^{18} + 222 x^{17} - 574 x^{16} - 2030 x^{15} + 11793 x^{14} + 4334 x^{13} - 104402 x^{12} + 38920 x^{11} + 512568 x^{10} - 267232 x^{9} - 1454928 x^{8} + 576352 x^{7} + 2268740 x^{6} - 159284 x^{5} - 1691906 x^{4} - 756560 x^{3} + 346947 x^{2} + 508220 x + 7775 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2106183370512833937071945009682254848=2^{10}\cdot 19^{12}\cdot 43^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.49$
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:  $2, 19, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{10} a^{15} - \frac{1}{2} a^{14} + \frac{1}{10} a^{13} + \frac{1}{10} a^{9} + \frac{1}{5} a^{8} - \frac{1}{2} a^{7} - \frac{3}{10} a^{6} - \frac{3}{10} a^{5} - \frac{3}{10} a^{4} - \frac{1}{2} a^{3} + \frac{1}{10} a - \frac{1}{2}$, $\frac{1}{10} a^{16} - \frac{2}{5} a^{14} - \frac{1}{2} a^{13} + \frac{1}{10} a^{10} - \frac{3}{10} a^{9} - \frac{1}{2} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{2} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{17} - \frac{1}{2} a^{14} + \frac{2}{5} a^{13} + \frac{1}{10} a^{11} - \frac{3}{10} a^{10} - \frac{1}{10} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{5} + \frac{3}{10} a^{4} + \frac{1}{10} a^{3} - \frac{1}{10} a$, $\frac{1}{10} a^{18} - \frac{1}{10} a^{14} - \frac{1}{2} a^{13} + \frac{1}{10} a^{12} - \frac{3}{10} a^{11} - \frac{1}{10} a^{10} - \frac{1}{2} a^{9} + \frac{1}{5} a^{8} - \frac{1}{2} a^{7} + \frac{3}{10} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{2} a^{3} - \frac{1}{10} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{217362507230920908962610174397654527559931731972850} a^{19} + \frac{7475089270967597312072177522214636706889734073581}{217362507230920908962610174397654527559931731972850} a^{18} + \frac{9211074060979419797246990844442431807858132505489}{217362507230920908962610174397654527559931731972850} a^{17} - \frac{1265065139575692267584820934090559413942371520627}{217362507230920908962610174397654527559931731972850} a^{16} - \frac{377889021295649081099168681668927913691719681427}{8360096431958496498561929784525174136920451229725} a^{15} + \frac{53841358233862593925097108953015728260148712120796}{108681253615460454481305087198827263779965865986425} a^{14} - \frac{2618672685233544080585466240000638389057534094954}{8360096431958496498561929784525174136920451229725} a^{13} + \frac{35152249512764381155839506448460149202627467803549}{108681253615460454481305087198827263779965865986425} a^{12} + \frac{2959462762984974835618071778730819782059998610473}{6210357349454883113217433554218700787426620913510} a^{11} + \frac{1456506531628331901729486842532267205149960136632}{3105178674727441556608716777109350393713310456755} a^{10} - \frac{6913676078738510044973440484378600723849733514798}{15525893373637207783043583885546751968566552283775} a^{9} - \frac{734426453875299263447591107473539533945445579751}{3105178674727441556608716777109350393713310456755} a^{8} - \frac{71478519155971867468155852315279138981593490300063}{217362507230920908962610174397654527559931731972850} a^{7} + \frac{6092461012970143790836660288917615239722062410458}{21736250723092090896261017439765452755993173197285} a^{6} + \frac{1636309713761890900582534127631063501957420632831}{3344038572783398599424771913810069654768180491890} a^{5} + \frac{3027385328902700984363262283174778433512531294071}{217362507230920908962610174397654527559931731972850} a^{4} + \frac{38064790732979128522937350759039020588141915436813}{217362507230920908962610174397654527559931731972850} a^{3} + \frac{102009897543508092666920950769125927419633862890577}{217362507230920908962610174397654527559931731972850} a^{2} + \frac{7899588092499083116658155294208709042159925242769}{21736250723092090896261017439765452755993173197285} a + \frac{4071952758891116780413681639654584025925972342515}{8694500289236836358504406975906181102397269278914}$

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:  $15$
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:  \( 118116765904 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T423:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 160 conjugacy class representatives for t20n423 are not computed
Character table for t20n423 is not computed

Intermediate fields

\(\Q(\sqrt{817}) \), 5.5.667489.1 x5, 10.10.364007458703857.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.13$x^{10} - 15 x^{8} + 26 x^{6} - 22 x^{4} + 37 x^{2} - 59$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
$19$19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.4.3.1$x^{4} + 387$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$