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Properties

Label 20.0.14315153843...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 5^{20}\cdot 7^{8}\cdot 97$
Root discriminant $36.12$
Ramified primes $2, 5, 7, 97$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group 20T633

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![853, -6890, 23555, -45850, 80395, -125026, 90110, 29100, -56505, -17430, 36707, 70, -10325, 160, 2510, -372, -285, 60, 25, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 25*x^18 + 60*x^17 - 285*x^16 - 372*x^15 + 2510*x^14 + 160*x^13 - 10325*x^12 + 70*x^11 + 36707*x^10 - 17430*x^9 - 56505*x^8 + 29100*x^7 + 90110*x^6 - 125026*x^5 + 80395*x^4 - 45850*x^3 + 23555*x^2 - 6890*x + 853)
 
gp: K = bnfinit(x^20 - 10*x^19 + 25*x^18 + 60*x^17 - 285*x^16 - 372*x^15 + 2510*x^14 + 160*x^13 - 10325*x^12 + 70*x^11 + 36707*x^10 - 17430*x^9 - 56505*x^8 + 29100*x^7 + 90110*x^6 - 125026*x^5 + 80395*x^4 - 45850*x^3 + 23555*x^2 - 6890*x + 853, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 25 x^{18} + 60 x^{17} - 285 x^{16} - 372 x^{15} + 2510 x^{14} + 160 x^{13} - 10325 x^{12} + 70 x^{11} + 36707 x^{10} - 17430 x^{9} - 56505 x^{8} + 29100 x^{7} + 90110 x^{6} - 125026 x^{5} + 80395 x^{4} - 45850 x^{3} + 23555 x^{2} - 6890 x + 853 \)

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:  \(14315153843200000000000000000000=2^{28}\cdot 5^{20}\cdot 7^{8}\cdot 97\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.12$
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:  $2, 5, 7, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{15} a^{13} + \frac{1}{15} a^{12} + \frac{7}{15} a^{11} + \frac{1}{5} a^{10} - \frac{1}{3} a^{9} - \frac{2}{5} a^{8} + \frac{4}{15} a^{7} - \frac{2}{15} a^{6} - \frac{1}{5} a^{5} + \frac{7}{15} a^{3} + \frac{7}{15} a^{2} - \frac{1}{15} a + \frac{1}{15}$, $\frac{1}{15} a^{14} + \frac{1}{15} a^{12} - \frac{4}{15} a^{11} + \frac{2}{15} a^{10} + \frac{4}{15} a^{9} + \frac{4}{15} a^{7} - \frac{1}{15} a^{6} - \frac{2}{15} a^{5} - \frac{1}{5} a^{4} - \frac{1}{3} a^{3} + \frac{7}{15} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{75} a^{15} - \frac{2}{15} a^{12} + \frac{2}{15} a^{11} - \frac{19}{75} a^{10} - \frac{1}{15} a^{9} - \frac{1}{5} a^{8} - \frac{1}{3} a^{7} - \frac{2}{5} a^{6} + \frac{1}{3} a^{5} + \frac{2}{5} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{4}{15} a + \frac{8}{25}$, $\frac{1}{675} a^{16} + \frac{1}{675} a^{15} - \frac{2}{135} a^{14} - \frac{1}{45} a^{13} - \frac{1}{45} a^{12} - \frac{304}{675} a^{11} + \frac{241}{675} a^{10} - \frac{22}{135} a^{9} - \frac{32}{135} a^{8} + \frac{7}{135} a^{7} - \frac{14}{45} a^{6} + \frac{11}{45} a^{5} + \frac{2}{135} a^{4} + \frac{2}{15} a^{3} + \frac{8}{135} a^{2} - \frac{146}{675} a + \frac{154}{675}$, $\frac{1}{675} a^{17} - \frac{2}{675} a^{15} - \frac{1}{135} a^{14} + \frac{71}{675} a^{12} - \frac{8}{135} a^{11} - \frac{8}{75} a^{10} + \frac{26}{135} a^{9} + \frac{19}{45} a^{8} - \frac{4}{135} a^{7} + \frac{7}{45} a^{6} - \frac{31}{135} a^{5} - \frac{4}{27} a^{4} - \frac{2}{27} a^{3} + \frac{88}{225} a^{2} + \frac{17}{45} a - \frac{163}{675}$, $\frac{1}{5567372664332625} a^{18} - \frac{1}{618596962703625} a^{17} + \frac{2190109564522}{5567372664332625} a^{16} - \frac{17520876515972}{5567372664332625} a^{15} + \frac{1703429677973}{74231635524435} a^{14} + \frac{154631113337321}{5567372664332625} a^{13} + \frac{346521619241681}{5567372664332625} a^{12} - \frac{762808942318036}{1855790888110875} a^{11} - \frac{2731123916653097}{5567372664332625} a^{10} + \frac{11599423978543}{24743878508145} a^{9} - \frac{107521126798}{422570980215} a^{8} + \frac{144042263754248}{371158177622175} a^{7} + \frac{17707510598513}{1113474532866525} a^{6} - \frac{328319078884103}{1113474532866525} a^{5} + \frac{17870837067287}{35918533318275} a^{4} - \frac{272276263051877}{1855790888110875} a^{3} + \frac{905050118992433}{1855790888110875} a^{2} + \frac{2019554528230163}{5567372664332625} a + \frac{680014019794879}{1855790888110875}$, $\frac{1}{228262279237637625} a^{19} + \frac{11}{228262279237637625} a^{18} + \frac{125909502105067}{228262279237637625} a^{17} + \frac{158248666817908}{228262279237637625} a^{16} - \frac{17593340132336}{15217485282509175} a^{15} - \frac{6280500227662529}{228262279237637625} a^{14} - \frac{1740404530620883}{76087426412545875} a^{13} - \frac{12307551220199813}{228262279237637625} a^{12} - \frac{9901401449678213}{25362475470848625} a^{11} - \frac{53003848665511}{298382064362925} a^{10} + \frac{1321908165081704}{9130491169505505} a^{9} + \frac{19493145998147714}{45652455847527525} a^{8} + \frac{7533098298480581}{15217485282509175} a^{7} - \frac{11529560130217558}{45652455847527525} a^{6} - \frac{5715459086498423}{45652455847527525} a^{5} + \frac{347553631279847}{8454158490282875} a^{4} - \frac{7933153526961544}{25362475470848625} a^{3} - \frac{19798770051771769}{76087426412545875} a^{2} - \frac{3462474001697354}{13427192896331625} a - \frac{8451222200929462}{45652455847527525}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{5}$, which has order $5$ (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:  \( \frac{1479763590}{1649591900543} a^{18} - \frac{13317872310}{1649591900543} a^{17} + \frac{23298572975}{1649591900543} a^{16} + \frac{115483188560}{1649591900543} a^{15} - \frac{312198852565}{1649591900543} a^{14} - \frac{892115035105}{1649591900543} a^{13} + \frac{2901640192185}{1649591900543} a^{12} + \frac{3366472423415}{1649591900543} a^{11} - \frac{12655076968819}{1649591900543} a^{10} - \frac{13399683205815}{1649591900543} a^{9} + \frac{83770675855}{3130155409} a^{8} + \frac{21682791773925}{1649591900543} a^{7} - \frac{73208211769845}{1649591900543} a^{6} - \frac{34978501894671}{1649591900543} a^{5} + \frac{3764170652480}{53212641953} a^{4} - \frac{61275037143615}{1649591900543} a^{3} + \frac{27960496195280}{1649591900543} a^{2} - \frac{20153955769650}{1649591900543} a + \frac{4557820464163}{1649591900543} \) (order $4$)
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:  \( 20344482.1317 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T633:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40960
The 124 conjugacy class representatives for t20n633 are not computed
Character table for t20n633 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.5.2450000.1, 10.0.384160000000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ $20$

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
2Data not computed
$5$5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.10.10.10$x^{10} + 10 x^{8} + 5 x^{6} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 2$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97Data not computed