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Properties

Label 20.0.15892878751...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{38}\cdot 3^{18}\cdot 5^{37}\cdot 29^{5}$
Root discriminant $457.15$
Ramified primes $2, 3, 5, 29$
Class number $40$ (GRH)
Class group $[2, 20]$ (GRH)
Galois group $D_4\times F_5$ (as 20T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11462824512, 36027296640, 25741488960, -11563309440, 2348943840, -2562120432, 614328480, 197617680, -56073540, -1729980, 2750173, -1837930, 877605, -236400, 59130, -15852, 3870, -660, 105, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 105*x^18 - 660*x^17 + 3870*x^16 - 15852*x^15 + 59130*x^14 - 236400*x^13 + 877605*x^12 - 1837930*x^11 + 2750173*x^10 - 1729980*x^9 - 56073540*x^8 + 197617680*x^7 + 614328480*x^6 - 2562120432*x^5 + 2348943840*x^4 - 11563309440*x^3 + 25741488960*x^2 + 36027296640*x + 11462824512)
 
gp: K = bnfinit(x^20 - 10*x^19 + 105*x^18 - 660*x^17 + 3870*x^16 - 15852*x^15 + 59130*x^14 - 236400*x^13 + 877605*x^12 - 1837930*x^11 + 2750173*x^10 - 1729980*x^9 - 56073540*x^8 + 197617680*x^7 + 614328480*x^6 - 2562120432*x^5 + 2348943840*x^4 - 11563309440*x^3 + 25741488960*x^2 + 36027296640*x + 11462824512, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 105 x^{18} - 660 x^{17} + 3870 x^{16} - 15852 x^{15} + 59130 x^{14} - 236400 x^{13} + 877605 x^{12} - 1837930 x^{11} + 2750173 x^{10} - 1729980 x^{9} - 56073540 x^{8} + 197617680 x^{7} + 614328480 x^{6} - 2562120432 x^{5} + 2348943840 x^{4} - 11563309440 x^{3} + 25741488960 x^{2} + 36027296640 x + 11462824512 \)

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:  \(158928787510637220000000000000000000000000000000000000=2^{38}\cdot 3^{18}\cdot 5^{37}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $457.15$
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:  $2, 3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{6} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{8} + \frac{1}{6} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{9} - \frac{1}{4} a^{5} + \frac{1}{6} a^{3}$, $\frac{1}{72} a^{10} + \frac{1}{72} a^{9} - \frac{1}{36} a^{8} + \frac{1}{36} a^{7} + \frac{5}{72} a^{6} - \frac{7}{72} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3}$, $\frac{1}{72} a^{11} - \frac{1}{24} a^{9} - \frac{1}{36} a^{8} + \frac{1}{24} a^{7} - \frac{17}{72} a^{5} + \frac{1}{12} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{144} a^{12} - \frac{1}{144} a^{10} - \frac{1}{24} a^{9} - \frac{1}{144} a^{8} + \frac{1}{36} a^{7} - \frac{7}{144} a^{6} + \frac{5}{72} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{144} a^{13} - \frac{1}{144} a^{11} + \frac{5}{144} a^{9} + \frac{1}{36} a^{8} + \frac{5}{144} a^{7} - \frac{1}{18} a^{6} + \frac{1}{24} a^{5} + \frac{1}{12} a^{4} - \frac{1}{6} a^{3}$, $\frac{1}{288} a^{14} - \frac{1}{288} a^{13} - \frac{1}{288} a^{12} - \frac{1}{288} a^{11} + \frac{1}{288} a^{10} - \frac{11}{288} a^{9} - \frac{11}{288} a^{8} - \frac{1}{96} a^{7} - \frac{1}{48} a^{6} + \frac{7}{36} a^{5} - \frac{1}{12} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{7200} a^{15} + \frac{1}{1440} a^{14} + \frac{1}{480} a^{13} - \frac{1}{480} a^{12} - \frac{7}{1440} a^{11} + \frac{31}{7200} a^{10} + \frac{17}{1440} a^{9} + \frac{31}{1440} a^{8} + \frac{23}{720} a^{7} - \frac{11}{180} a^{6} + \frac{31}{450} a^{5} + \frac{7}{60} a^{4} - \frac{9}{20} a^{3} - \frac{3}{10} a^{2} + \frac{2}{5} a - \frac{3}{25}$, $\frac{1}{21600} a^{16} + \frac{1}{21600} a^{15} - \frac{1}{4320} a^{14} + \frac{1}{864} a^{13} + \frac{1}{864} a^{12} - \frac{29}{21600} a^{11} + \frac{61}{21600} a^{10} + \frac{23}{4320} a^{9} + \frac{61}{2160} a^{8} - \frac{2}{45} a^{7} - \frac{137}{1800} a^{6} - \frac{1}{150} a^{5} - \frac{1}{12} a^{4} - \frac{1}{6} a^{3} - \frac{3}{10} a^{2} - \frac{6}{25} a + \frac{4}{25}$, $\frac{1}{21600} a^{17} - \frac{1}{1440} a^{14} + \frac{1}{1440} a^{13} - \frac{23}{7200} a^{12} + \frac{7}{1440} a^{11} - \frac{1}{160} a^{10} + \frac{13}{360} a^{9} + \frac{37}{4320} a^{8} - \frac{143}{7200} a^{7} - \frac{11}{240} a^{6} - \frac{7}{90} a^{5} - \frac{11}{60} a^{4} + \frac{13}{60} a^{3} - \frac{29}{100} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{6574953830382949005172800} a^{18} + \frac{49085543705602480417}{2191651276794316335057600} a^{17} + \frac{13207076279225656451}{3287476915191474502586400} a^{16} - \frac{13796117771911432891}{205467307199467156411650} a^{15} + \frac{1103975157103799436127}{657495383038294900517280} a^{14} + \frac{7412385919388023343993}{3287476915191474502586400} a^{13} + \frac{1282742655875097250981}{410934614398934312823300} a^{12} + \frac{1635204299889554997013}{1643738457595737251293200} a^{11} - \frac{26701536453752029591727}{6574953830382949005172800} a^{10} - \frac{13793757889482817553483}{438330255358863267011520} a^{9} - \frac{83188998301993590217747}{3287476915191474502586400} a^{8} - \frac{3301366093613154479419}{182637606399526361254800} a^{7} + \frac{561241495954996553327}{11414850399970397578425} a^{6} - \frac{20223338070491373276631}{91318803199763180627400} a^{5} - \frac{3465796107781142967539}{18263760639952636125480} a^{4} + \frac{9059603638371165683827}{22829700799940795156850} a^{3} + \frac{1564796754389936552662}{3804950133323465859475} a^{2} + \frac{99664651643533337989}{3804950133323465859475} a + \frac{1582326562243045258261}{3804950133323465859475}$, $\frac{1}{7379065021491225041448133497571556806814555304000} a^{19} + \frac{270635449056297471008659}{3689532510745612520724066748785778403407277652000} a^{18} - \frac{39246110468287382538268534732876480306562651}{7379065021491225041448133497571556806814555304000} a^{17} - \frac{210345868225066372780237654563373533628577}{307461042562134376727005562398814866950606471000} a^{16} - \frac{21950818818442992198026169837121270156641799}{1229844170248537506908022249595259467802425884000} a^{15} - \frac{1351498056304174290616011307695817597662046529}{1229844170248537506908022249595259467802425884000} a^{14} + \frac{153461235262362619922925524947330458604649117}{307461042562134376727005562398814866950606471000} a^{13} - \frac{2013622361723124967524319952189860988128450031}{1229844170248537506908022249595259467802425884000} a^{12} + \frac{10287889939151164373155724144196085884986917129}{2459688340497075013816044499190518935604851768000} a^{11} - \frac{1509976670888593816620536250052768307529654083}{922383127686403130181016687196444600851819413000} a^{10} - \frac{24256465148735109855446052128154678253439759909}{7379065021491225041448133497571556806814555304000} a^{9} + \frac{146393585628323936944021158516774293048328529609}{3689532510745612520724066748785778403407277652000} a^{8} + \frac{5010335928735810281169856570302629389178352001}{136649352249837500767558027732806607533602876000} a^{7} - \frac{543131722598889005084048083197661841579731118}{12810876773422265696958565099950619456275269625} a^{6} + \frac{532411013539644049639843534818162384726744177}{4270292257807421898986188366650206485425089875} a^{5} - \frac{12712009279000243607819654353675695097910046907}{102487014187378125575668520799604955650202157000} a^{4} + \frac{1954021525301089669449518584556888230624282676}{4270292257807421898986188366650206485425089875} a^{3} - \frac{4709798344614189545998889578747796133841197263}{17081169031229687595944753466600825941700359500} a^{2} + \frac{1915115720934411366143703787868328893464524943}{8540584515614843797972376733300412970850179750} a - \frac{2021980027701918530265225644338956822792678083}{4270292257807421898986188366650206485425089875}$

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

Class group and class number

$C_{2}\times C_{20}$, which has order $40$ (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:  \( -1 \) (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:  \( 664049058604952300 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times F_5$ (as 20T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 25 conjugacy class representatives for $D_4\times F_5$
Character table for $D_4\times F_5$ is not computed

Intermediate fields

\(\Q(\sqrt{6}) \), 4.0.83520.3, 5.1.2531250000.14, 10.2.39366000000000000000000.22

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ R $20$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.19.49$x^{10} - 6$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
2.10.19.49$x^{10} - 6$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$3$3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.5.9.5$x^{5} + 105$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.5$x^{5} + 105$$5$$1$$9$$F_5$$[9/4]_{4}$
5.10.19.4$x^{10} + 105$$10$$1$$19$$F_5$$[9/4]_{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$