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Properties

Label 20.20.6096448022...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{48}\cdot 5^{20}\cdot 17^{6}\cdot 97^{2}$
Root discriminant $97.56$
Ramified primes $2, 5, 17, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T872

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![514706, -327680, -4563040, 4761040, 9821380, -10171560, -10571900, 8539280, 6840010, -3253820, -2506780, 581720, 490950, -52020, -52700, 2280, 3080, -40, -90, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 90*x^18 - 40*x^17 + 3080*x^16 + 2280*x^15 - 52700*x^14 - 52020*x^13 + 490950*x^12 + 581720*x^11 - 2506780*x^10 - 3253820*x^9 + 6840010*x^8 + 8539280*x^7 - 10571900*x^6 - 10171560*x^5 + 9821380*x^4 + 4761040*x^3 - 4563040*x^2 - 327680*x + 514706)
 
gp: K = bnfinit(x^20 - 90*x^18 - 40*x^17 + 3080*x^16 + 2280*x^15 - 52700*x^14 - 52020*x^13 + 490950*x^12 + 581720*x^11 - 2506780*x^10 - 3253820*x^9 + 6840010*x^8 + 8539280*x^7 - 10571900*x^6 - 10171560*x^5 + 9821380*x^4 + 4761040*x^3 - 4563040*x^2 - 327680*x + 514706, 1)
 

Normalized defining polynomial

\( x^{20} - 90 x^{18} - 40 x^{17} + 3080 x^{16} + 2280 x^{15} - 52700 x^{14} - 52020 x^{13} + 490950 x^{12} + 581720 x^{11} - 2506780 x^{10} - 3253820 x^{9} + 6840010 x^{8} + 8539280 x^{7} - 10571900 x^{6} - 10171560 x^{5} + 9821380 x^{4} + 4761040 x^{3} - 4563040 x^{2} - 327680 x + 514706 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6096448022178797977600000000000000000000=2^{48}\cdot 5^{20}\cdot 17^{6}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.56$
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:  $2, 5, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{24031299311} a^{18} + \frac{6536544811}{24031299311} a^{17} + \frac{11084055977}{24031299311} a^{16} + \frac{11062511493}{24031299311} a^{15} + \frac{1217927340}{24031299311} a^{14} - \frac{918666959}{24031299311} a^{13} + \frac{1968036015}{24031299311} a^{12} - \frac{9790154258}{24031299311} a^{11} - \frac{11505603415}{24031299311} a^{10} + \frac{1691705707}{24031299311} a^{9} - \frac{9898275436}{24031299311} a^{8} + \frac{42615466}{125818321} a^{7} - \frac{2512506308}{24031299311} a^{6} - \frac{1483841815}{24031299311} a^{5} + \frac{3764817155}{24031299311} a^{4} + \frac{8463059843}{24031299311} a^{3} + \frac{7674278839}{24031299311} a^{2} + \frac{10086782346}{24031299311} a + \frac{10667771668}{24031299311}$, $\frac{1}{22943184806069751967490412269007506450989} a^{19} + \frac{111723485406278485792568417893}{22943184806069751967490412269007506450989} a^{18} - \frac{2321093845264734535328373110895748293020}{22943184806069751967490412269007506450989} a^{17} + \frac{186975436643422668272828629615285396710}{22943184806069751967490412269007506450989} a^{16} + \frac{7541971620794508998820092018340190079152}{22943184806069751967490412269007506450989} a^{15} + \frac{2933783034014420906786480823475352261437}{22943184806069751967490412269007506450989} a^{14} - \frac{6770607690672557528556020736669184393893}{22943184806069751967490412269007506450989} a^{13} + \frac{7116779978296972329265468739693963114932}{22943184806069751967490412269007506450989} a^{12} + \frac{6442630339659370613403258653472355501648}{22943184806069751967490412269007506450989} a^{11} - \frac{10590116244311418330371790346718374020596}{22943184806069751967490412269007506450989} a^{10} - \frac{6500117813527227883777119543055726455731}{22943184806069751967490412269007506450989} a^{9} + \frac{6316723465956452323870609660609329387309}{22943184806069751967490412269007506450989} a^{8} - \frac{8843913955738603340692968121993582476631}{22943184806069751967490412269007506450989} a^{7} + \frac{4334282428249816001786342002700079702231}{22943184806069751967490412269007506450989} a^{6} + \frac{10613156800077425768666006541801149343733}{22943184806069751967490412269007506450989} a^{5} + \frac{7824972404882111098311288329534352425773}{22943184806069751967490412269007506450989} a^{4} + \frac{7907844849952388901094100531239498801145}{22943184806069751967490412269007506450989} a^{3} - \frac{3633092716261577193198004077545573665516}{22943184806069751967490412269007506450989} a^{2} - \frac{3946256417227633587745031978241410643248}{22943184806069751967490412269007506450989} a + \frac{5258601076596262779250770297083270104715}{22943184806069751967490412269007506450989}$

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:  $19$
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:  \( 65803151167000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T872:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 204800
The 116 conjugacy class representatives for t20n872 are not computed
Character table for t20n872 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.1479680000000000.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.10.14$x^{10} + 10 x^{6} + 10 x^{5} + 100 x^{2} + 50 x + 25$$5$$2$$10$$(C_5^2 : C_4) : C_2$$[5/4, 5/4]_{4}^{2}$
5.10.10.14$x^{10} + 10 x^{6} + 10 x^{5} + 100 x^{2} + 50 x + 25$$5$$2$$10$$(C_5^2 : C_4) : C_2$$[5/4, 5/4]_{4}^{2}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{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$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$