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Properties

Label 20.20.1389549804...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{40}\cdot 5^{10}\cdot 13^{4}\cdot 29^{4}\cdot 31^{4}\cdot 1381^{6}$
Root discriminant $509.50$
Ramified primes $2, 5, 13, 29, 31, 1381$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28413322415166317694976, 0, -2273477282314176759808, 0, 73545110298746947584, 0, -1288401536737102336, 0, 13732302535071360, 0, -93996224024960, 0, 421312835904, 0, -1226506384, 0, 2225780, 0, -2278, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2278*x^18 + 2225780*x^16 - 1226506384*x^14 + 421312835904*x^12 - 93996224024960*x^10 + 13732302535071360*x^8 - 1288401536737102336*x^6 + 73545110298746947584*x^4 - 2273477282314176759808*x^2 + 28413322415166317694976)
 
gp: K = bnfinit(x^20 - 2278*x^18 + 2225780*x^16 - 1226506384*x^14 + 421312835904*x^12 - 93996224024960*x^10 + 13732302535071360*x^8 - 1288401536737102336*x^6 + 73545110298746947584*x^4 - 2273477282314176759808*x^2 + 28413322415166317694976, 1)
 

Normalized defining polynomial

\( x^{20} - 2278 x^{18} + 2225780 x^{16} - 1226506384 x^{14} + 421312835904 x^{12} - 93996224024960 x^{10} + 13732302535071360 x^{8} - 1288401536737102336 x^{6} + 73545110298746947584 x^{4} - 2273477282314176759808 x^{2} + 28413322415166317694976 \)

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:  \(1389549804808197507073369468751841433841827840000000000=2^{40}\cdot 5^{10}\cdot 13^{4}\cdot 29^{4}\cdot 31^{4}\cdot 1381^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $509.50$
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:  $2, 5, 13, 29, 31, 1381$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{32} a^{9} - \frac{1}{16} a^{7} - \frac{1}{8} a^{5}$, $\frac{1}{44192} a^{10} + \frac{121}{11048} a^{8} - \frac{49}{5524} a^{6} - \frac{101}{2762} a^{4} + \frac{232}{1381} a^{2}$, $\frac{1}{88384} a^{11} + \frac{121}{22096} a^{9} - \frac{49}{11048} a^{7} + \frac{1179}{11048} a^{5} + \frac{116}{1381} a^{3}$, $\frac{1}{244116608} a^{12} - \frac{1139}{122058304} a^{10} + \frac{556445}{61029152} a^{8} - \frac{370209}{15257288} a^{6} + \frac{893739}{7628644} a^{4} + \frac{243}{2762} a^{2}$, $\frac{1}{244116608} a^{13} + \frac{121}{61029152} a^{11} + \frac{890647}{61029152} a^{9} - \frac{218939}{7628644} a^{7} - \frac{398645}{15257288} a^{5} + \frac{475}{2762} a^{3}$, $\frac{1}{674250071296} a^{14} + \frac{121}{168562517824} a^{12} - \frac{508257}{84281258912} a^{10} + \frac{615137247}{84281258912} a^{8} - \frac{128544803}{5267578682} a^{6} - \frac{157887}{3814322} a^{4} - \frac{183}{1381} a^{2}$, $\frac{1}{674250071296} a^{15} + \frac{121}{168562517824} a^{13} + \frac{890647}{168562517824} a^{11} + \frac{1076670209}{84281258912} a^{9} - \frac{607630101}{21070314728} a^{7} + \frac{996651}{15257288} a^{5} - \frac{67}{1381} a^{3}$, $\frac{1}{1862278696919552} a^{16} + \frac{121}{465569674229888} a^{14} + \frac{890647}{465569674229888} a^{12} - \frac{389279783}{58196209278736} a^{10} + \frac{875836590351}{116392418557472} a^{8} + \frac{1056111939}{21070314728} a^{6} + \frac{314801}{3814322} a^{4} + \frac{36}{1381} a^{2}$, $\frac{1}{3724557393839104} a^{17} - \frac{1139}{1862278696919552} a^{15} + \frac{556445}{931139348459776} a^{13} - \frac{76656649}{232784837114944} a^{11} + \frac{6583013061}{58196209278736} a^{9} - \frac{531749095}{21070314728} a^{7} - \frac{961785}{15257288} a^{5} + \frac{219}{2762} a^{3}$, $\frac{1}{2893590723560933465196952188600445053597940736} a^{18} - \frac{33876866939529182931855423603}{1446795361780466732598476094300222526798970368} a^{16} + \frac{385862619448063729862279276309127}{723397680890233366299238047150111263399485184} a^{14} + \frac{169481159557681482384634111642304393}{180849420222558341574809511787527815849871296} a^{12} + \frac{281518003032929029494168227180672805081}{90424710111279170787404755893763907924935648} a^{10} + \frac{3538015457970435579582142940276379185}{120363427651636669383072735940713128707} a^{8} - \frac{614293316788335947850452536699977205}{11853313657221279533742137645139019192} a^{6} - \frac{444093405299007412574799338033611}{4291569028682577673331693571737516} a^{4} + \frac{215273332066837389919604611809}{1553790379682323560221467621918} a^{2} - \frac{225054671685242940463250811}{562559876785779710434999139}$, $\frac{1}{5787181447121866930393904377200890107195881472} a^{19} - \frac{33876866939529182931855423603}{2893590723560933465196952188600445053597940736} a^{17} + \frac{385862619448063729862279276309127}{1446795361780466732598476094300222526798970368} a^{15} + \frac{169481159557681482384634111642304393}{361698840445116683149619023575055631699742592} a^{13} + \frac{281518003032929029494168227180672805081}{180849420222558341574809511787527815849871296} a^{11} + \frac{3538015457970435579582142940276379185}{240726855303273338766145471881426257414} a^{9} + \frac{433685445182161996933657334471200097}{11853313657221279533742137645139019192} a^{7} + \frac{78599856483954625719765506862596}{1072892257170644418332923392934379} a^{5} - \frac{280810928887162195095564599575}{1553790379682323560221467621918} a^{3} + \frac{168752602550268384985874164}{562559876785779710434999139} a$

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:  \( 800664910826000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.109268775200000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: 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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
2Data not computed
5Data not computed
$13$13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
29.10.0.1$x^{10} + x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.8.0.1$x^{8} - x + 22$$1$$8$$0$$C_8$$[\ ]^{8}$
1381Data not computed