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Properties

Label 20.12.1882282076...3328.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{42}\cdot 97^{5}\cdot 2657^{4}$
Root discriminant $65.12$
Ramified primes $2, 97, 2657$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T658

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![472392, 314928, -2991816, 1329696, 3432132, -2227824, -1491372, 998784, 204210, 12360, 83872, -110868, -55312, 20284, 10482, -392, -536, -64, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 20*x^18 - 64*x^17 - 536*x^16 - 392*x^15 + 10482*x^14 + 20284*x^13 - 55312*x^12 - 110868*x^11 + 83872*x^10 + 12360*x^9 + 204210*x^8 + 998784*x^7 - 1491372*x^6 - 2227824*x^5 + 3432132*x^4 + 1329696*x^3 - 2991816*x^2 + 314928*x + 472392)
 
gp: K = bnfinit(x^20 - 20*x^18 - 64*x^17 - 536*x^16 - 392*x^15 + 10482*x^14 + 20284*x^13 - 55312*x^12 - 110868*x^11 + 83872*x^10 + 12360*x^9 + 204210*x^8 + 998784*x^7 - 1491372*x^6 - 2227824*x^5 + 3432132*x^4 + 1329696*x^3 - 2991816*x^2 + 314928*x + 472392, 1)
 

Normalized defining polynomial

\( x^{20} - 20 x^{18} - 64 x^{17} - 536 x^{16} - 392 x^{15} + 10482 x^{14} + 20284 x^{13} - 55312 x^{12} - 110868 x^{11} + 83872 x^{10} + 12360 x^{9} + 204210 x^{8} + 998784 x^{7} - 1491372 x^{6} - 2227824 x^{5} + 3432132 x^{4} + 1329696 x^{3} - 2991816 x^{2} + 314928 x + 472392 \)

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:  \(1882282076712230748707114352092643328=2^{42}\cdot 97^{5}\cdot 2657^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.12$
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:  $2, 97, 2657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{18} a^{12} - \frac{1}{9} a^{10} + \frac{4}{9} a^{9} + \frac{2}{9} a^{8} + \frac{2}{9} a^{7} + \frac{1}{3} a^{6} - \frac{1}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{3} a^{3} - \frac{4}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{54} a^{13} - \frac{1}{27} a^{11} - \frac{5}{27} a^{10} + \frac{11}{27} a^{9} + \frac{11}{27} a^{8} + \frac{4}{9} a^{7} - \frac{1}{27} a^{6} - \frac{8}{27} a^{5} + \frac{2}{9} a^{4} - \frac{4}{27} a^{3} - \frac{1}{9} a^{2}$, $\frac{1}{162} a^{14} - \frac{1}{81} a^{12} - \frac{5}{81} a^{11} + \frac{38}{81} a^{10} - \frac{16}{81} a^{9} - \frac{5}{27} a^{8} - \frac{1}{81} a^{7} - \frac{35}{81} a^{6} + \frac{11}{27} a^{5} + \frac{23}{81} a^{4} + \frac{8}{27} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{486} a^{15} - \frac{1}{243} a^{13} - \frac{5}{243} a^{12} + \frac{38}{243} a^{11} - \frac{97}{243} a^{10} - \frac{32}{81} a^{9} - \frac{82}{243} a^{8} - \frac{35}{243} a^{7} - \frac{16}{81} a^{6} - \frac{58}{243} a^{5} + \frac{35}{81} a^{4} - \frac{1}{9} a^{3}$, $\frac{1}{2916} a^{16} - \frac{1}{1458} a^{14} - \frac{5}{1458} a^{13} + \frac{19}{729} a^{12} + \frac{73}{729} a^{11} + \frac{211}{486} a^{10} + \frac{202}{729} a^{9} - \frac{139}{729} a^{8} - \frac{89}{243} a^{7} - \frac{29}{729} a^{6} - \frac{23}{243} a^{5} + \frac{17}{54} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{8748} a^{17} - \frac{1}{4374} a^{15} - \frac{5}{4374} a^{14} + \frac{19}{2187} a^{13} - \frac{97}{4374} a^{12} + \frac{211}{1458} a^{11} - \frac{1013}{2187} a^{10} + \frac{347}{2187} a^{9} - \frac{251}{729} a^{8} + \frac{943}{2187} a^{7} - \frac{23}{729} a^{6} + \frac{35}{162} a^{5} - \frac{1}{9} a^{4} - \frac{2}{9} a^{3} + \frac{1}{9} a^{2}$, $\frac{1}{26244} a^{18} - \frac{1}{13122} a^{16} - \frac{5}{13122} a^{15} + \frac{19}{6561} a^{14} - \frac{97}{13122} a^{13} - \frac{16}{2187} a^{12} - \frac{1013}{6561} a^{11} - \frac{1111}{6561} a^{10} + \frac{964}{2187} a^{9} - \frac{2702}{6561} a^{8} + \frac{949}{2187} a^{7} + \frac{197}{486} a^{6} - \frac{7}{27} a^{5} - \frac{5}{27} a^{4} + \frac{10}{27} a^{3} + \frac{1}{9} a^{2}$, $\frac{1}{264667769419273118658733888345965928301523708} a^{19} + \frac{351318292389461133444426495779756414945}{44111294903212186443122314724327654716920618} a^{18} - \frac{5060374936531781238088223037508718854283}{264667769419273118658733888345965928301523708} a^{17} + \frac{25386259489859799592526709737772126326759}{264667769419273118658733888345965928301523708} a^{16} - \frac{44446085095866970381713968956690961691749}{66166942354818279664683472086491482075380927} a^{15} - \frac{47531369963211714649938494247432426096368}{66166942354818279664683472086491482075380927} a^{14} - \frac{19274406704279381806748718083752819709413}{2450627494622899246840128595795980817606701} a^{13} - \frac{798106695538245724979338972522921786109246}{66166942354818279664683472086491482075380927} a^{12} - \frac{2358859640769004377670649541933690059738993}{132333884709636559329366944172982964150761854} a^{11} + \frac{5124392861646566449766099153185041313255}{14703764967737395481040771574775884905640206} a^{10} - \frac{8595859721350466045101920479170893808589342}{66166942354818279664683472086491482075380927} a^{9} - \frac{346631308590216828958290055840639726116220}{22055647451606093221561157362163827358460309} a^{8} - \frac{2546177211621472759161163911167269871297333}{14703764967737395481040771574775884905640206} a^{7} - \frac{299833280285727767179529437039368510173816}{2450627494622899246840128595795980817606701} a^{6} - \frac{35253839836607722270345429141728226719313}{181527962564659203469639155244146727230126} a^{5} - \frac{7502386342127774002721843514079774080329}{20169773618295467052182128360460747470014} a^{4} + \frac{9778764716231970352927456862126328408154}{30254660427443200578273192540691121205021} a^{3} + \frac{9092120454552743997499612683434100985891}{30254660427443200578273192540691121205021} a^{2} + \frac{481211860856993423760492503925628574123}{10084886809147733526091064180230373735007} a + \frac{405368566140195546808977267279201162980}{3361628936382577842030354726743457911669}$

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:  \( 151968204084 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T658:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 76 conjugacy class representatives for t20n658 are not computed
Character table for t20n658 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.24832.1, 10.6.925322313728.1

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 24 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.12.26.64$x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{6} - 2 x^{4} + 4 x^{3} + 2$$12$$1$$26$$S_3 \times C_2^2$$[2, 3]_{3}^{2}$
$97$97.2.1.2$x^{2} + 485$$2$$1$$1$$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}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
2657Data not computed