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Properties

Label 20.8.97339589945...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{30}\cdot 5^{5}\cdot 11^{10}\cdot 5783^{4}$
Root discriminant $79.33$
Ramified primes $2, 5, 11, 5783$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T466

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-56371, -497984, -844676, 1671184, 1984585, -3024628, -2288088, 2499778, 2087755, -399580, -810376, -275092, 27931, 52536, 7804, -3954, -648, 126, -20, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 20*x^18 + 126*x^17 - 648*x^16 - 3954*x^15 + 7804*x^14 + 52536*x^13 + 27931*x^12 - 275092*x^11 - 810376*x^10 - 399580*x^9 + 2087755*x^8 + 2499778*x^7 - 2288088*x^6 - 3024628*x^5 + 1984585*x^4 + 1671184*x^3 - 844676*x^2 - 497984*x - 56371)
 
gp: K = bnfinit(x^20 - 6*x^19 - 20*x^18 + 126*x^17 - 648*x^16 - 3954*x^15 + 7804*x^14 + 52536*x^13 + 27931*x^12 - 275092*x^11 - 810376*x^10 - 399580*x^9 + 2087755*x^8 + 2499778*x^7 - 2288088*x^6 - 3024628*x^5 + 1984585*x^4 + 1671184*x^3 - 844676*x^2 - 497984*x - 56371, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 20 x^{18} + 126 x^{17} - 648 x^{16} - 3954 x^{15} + 7804 x^{14} + 52536 x^{13} + 27931 x^{12} - 275092 x^{11} - 810376 x^{10} - 399580 x^{9} + 2087755 x^{8} + 2499778 x^{7} - 2288088 x^{6} - 3024628 x^{5} + 1984585 x^{4} + 1671184 x^{3} - 844676 x^{2} - 497984 x - 56371 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(97339589945843039759666267763507200000=2^{30}\cdot 5^{5}\cdot 11^{10}\cdot 5783^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.33$
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:  $2, 5, 11, 5783$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{11} a^{16} - \frac{4}{11} a^{15} - \frac{4}{11} a^{13} - \frac{5}{11} a^{12} - \frac{1}{11} a^{11} + \frac{1}{11} a^{10} + \frac{3}{11} a^{9} - \frac{3}{11} a^{8} + \frac{5}{11} a^{7} + \frac{5}{11} a^{6} - \frac{5}{11} a^{4} - \frac{4}{11} a^{3} - \frac{3}{11} a^{2} + \frac{3}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{17} - \frac{5}{11} a^{15} - \frac{4}{11} a^{14} + \frac{1}{11} a^{13} + \frac{1}{11} a^{12} - \frac{3}{11} a^{11} - \frac{4}{11} a^{10} - \frac{2}{11} a^{9} + \frac{4}{11} a^{8} + \frac{3}{11} a^{7} - \frac{2}{11} a^{6} - \frac{5}{11} a^{5} - \frac{2}{11} a^{4} + \frac{3}{11} a^{3} + \frac{2}{11} a^{2} + \frac{2}{11} a + \frac{4}{11}$, $\frac{1}{11} a^{18} - \frac{2}{11} a^{15} + \frac{1}{11} a^{14} + \frac{3}{11} a^{13} + \frac{5}{11} a^{12} + \frac{2}{11} a^{11} + \frac{3}{11} a^{10} - \frac{3}{11} a^{9} - \frac{1}{11} a^{8} + \frac{1}{11} a^{7} - \frac{2}{11} a^{6} - \frac{2}{11} a^{5} + \frac{4}{11} a^{3} - \frac{2}{11} a^{2} - \frac{3}{11} a + \frac{5}{11}$, $\frac{1}{55646668560540894184064348231351792745677713198015146234517563043} a^{19} + \frac{1620120021124512543396431566421304661745200426475206717744265373}{55646668560540894184064348231351792745677713198015146234517563043} a^{18} - \frac{651660745573769265020043473225073413613285756511974730884128335}{55646668560540894184064348231351792745677713198015146234517563043} a^{17} + \frac{407653544043294637332107805475071647554958633429640723400454784}{55646668560540894184064348231351792745677713198015146234517563043} a^{16} + \frac{1667024077077841724006210725842625337136366256244101406638687524}{5058788050958263107642213475577435704152519381637740566774323913} a^{15} + \frac{603712097698694384689325108668246663505928791966775757333349576}{1918850640018651523588425801081096301575093558552246421879915967} a^{14} + \frac{22674380196527218711474372110610043545520283673945977176113914218}{55646668560540894184064348231351792745677713198015146234517563043} a^{13} + \frac{10980663524688623492728630058488418972862551962307306205277205860}{55646668560540894184064348231351792745677713198015146234517563043} a^{12} + \frac{837344355498740243951827897952056869491865056477767257058687062}{55646668560540894184064348231351792745677713198015146234517563043} a^{11} + \frac{24163783089389614054064780466132572924859710739543250571884320977}{55646668560540894184064348231351792745677713198015146234517563043} a^{10} + \frac{10188777951591479313762156315042586471412776950816099097484278580}{55646668560540894184064348231351792745677713198015146234517563043} a^{9} + \frac{2227545920667815461952603725800757929766239162450578501316860884}{5058788050958263107642213475577435704152519381637740566774323913} a^{8} - \frac{10464557538178432904562730697776281030501875473891024997624352744}{55646668560540894184064348231351792745677713198015146234517563043} a^{7} - \frac{1163102615538415572621390098785672140210846197097747663616694847}{5058788050958263107642213475577435704152519381637740566774323913} a^{6} - \frac{20662003907709224714595993150021665076453039956810765141589709738}{55646668560540894184064348231351792745677713198015146234517563043} a^{5} + \frac{17580781782933534025082211480024640766486562646570577292611906869}{55646668560540894184064348231351792745677713198015146234517563043} a^{4} - \frac{1689675616592744216703070768230768808764714394984899988644846306}{55646668560540894184064348231351792745677713198015146234517563043} a^{3} + \frac{23875083694530514565772495349204407540762253642964300084862878379}{55646668560540894184064348231351792745677713198015146234517563043} a^{2} - \frac{5515582206622170626387297485639735430605011354339561406014323353}{55646668560540894184064348231351792745677713198015146234517563043} a - \frac{212306968007911442519749528826075182295283305103078183689672147}{55646668560540894184064348231351792745677713198015146234517563043}$

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:  $13$
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:  \( 222619311064 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), 5.3.5783.1, 10.6.5515307956775936.1

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 $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ $20$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
2Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5783Data not computed