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Properties

Label 20.0.12806691984...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{22}\cdot 31^{10}$
Root discriminant $56.93$
Ramified primes $2, 5, 31$
Class number $30$ (GRH)
Class group $[30]$ (GRH)
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![292051855, -14132000, 471698245, -11630730, 359925375, -898430, 157291620, 651530, 42616025, 141990, 7484049, 9510, 867255, 130, 65780, -4, 3135, 0, 85, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 85*x^18 + 3135*x^16 - 4*x^15 + 65780*x^14 + 130*x^13 + 867255*x^12 + 9510*x^11 + 7484049*x^10 + 141990*x^9 + 42616025*x^8 + 651530*x^7 + 157291620*x^6 - 898430*x^5 + 359925375*x^4 - 11630730*x^3 + 471698245*x^2 - 14132000*x + 292051855)
 
gp: K = bnfinit(x^20 + 85*x^18 + 3135*x^16 - 4*x^15 + 65780*x^14 + 130*x^13 + 867255*x^12 + 9510*x^11 + 7484049*x^10 + 141990*x^9 + 42616025*x^8 + 651530*x^7 + 157291620*x^6 - 898430*x^5 + 359925375*x^4 - 11630730*x^3 + 471698245*x^2 - 14132000*x + 292051855, 1)
 

Normalized defining polynomial

\( x^{20} + 85 x^{18} + 3135 x^{16} - 4 x^{15} + 65780 x^{14} + 130 x^{13} + 867255 x^{12} + 9510 x^{11} + 7484049 x^{10} + 141990 x^{9} + 42616025 x^{8} + 651530 x^{7} + 157291620 x^{6} - 898430 x^{5} + 359925375 x^{4} - 11630730 x^{3} + 471698245 x^{2} - 14132000 x + 292051855 \)

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:  \(128066919840750156250000000000000000=2^{16}\cdot 5^{22}\cdot 31^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.93$
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:  $2, 5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} + \frac{1}{7} a^{15} - \frac{3}{14} a^{13} + \frac{3}{14} a^{12} - \frac{3}{14} a^{11} - \frac{3}{14} a^{10} - \frac{3}{7} a^{9} + \frac{5}{14} a^{8} + \frac{5}{14} a^{7} + \frac{1}{7} a^{6} + \frac{5}{14} a^{5} + \frac{3}{14} a^{4} + \frac{3}{14} a^{3} + \frac{1}{14} a^{2} + \frac{3}{14} a + \frac{3}{7}$, $\frac{1}{13949321056260329000405502589745549650602011844947385996058017186} a^{19} + \frac{63981186351611159524897841492156810972496088807914773438856979}{1992760150894332714343643227106507092943144549278197999436859598} a^{18} + \frac{2662852328105233518758691858076834978176824934388492243490791331}{13949321056260329000405502589745549650602011844947385996058017186} a^{17} + \frac{2671914373327815724953552840015219458159259966519992175074925707}{13949321056260329000405502589745549650602011844947385996058017186} a^{16} + \frac{2777582463659816774420369506122595796490149324020002401065860779}{13949321056260329000405502589745549650602011844947385996058017186} a^{15} - \frac{221274286548361046292199829406470745930576045560581230705647753}{13949321056260329000405502589745549650602011844947385996058017186} a^{14} + \frac{197353851626454700652436252725609333103014514120284679455167292}{2324886842710054833400917098290924941767001974157897666009669531} a^{13} - \frac{1704695278294748580234698584158142399197265388894641731915823585}{6974660528130164500202751294872774825301005922473692998029008593} a^{12} - \frac{930773116540685855178983774253589428551816765772980001007821917}{13949321056260329000405502589745549650602011844947385996058017186} a^{11} - \frac{98123651908523947974246017321587483798728908757488586710008061}{13949321056260329000405502589745549650602011844947385996058017186} a^{10} + \frac{4273348961608180893921501134966756258756230529317396833837370909}{13949321056260329000405502589745549650602011844947385996058017186} a^{9} + \frac{2062032783600002714438681098490314659515043071998647696143101994}{6974660528130164500202751294872774825301005922473692998029008593} a^{8} - \frac{279941362876318227961409936902445871536744067268795417998232926}{774962280903351611133639032763641647255667324719299222003223177} a^{7} + \frac{1125883426404309860650180223552738881736416273883841836159152912}{6974660528130164500202751294872774825301005922473692998029008593} a^{6} - \frac{3304701900217632503485843984915006674747908577172813759724312519}{6974660528130164500202751294872774825301005922473692998029008593} a^{5} - \frac{340913380609371902422746167170305031248393406919478635413420823}{1549924561806703222267278065527283294511334649438598444006446354} a^{4} - \frac{1121081234359878572290818461364451919890385258667609507760383184}{2324886842710054833400917098290924941767001974157897666009669531} a^{3} + \frac{176567687466332933409188172364888265620635107523651016831386413}{774962280903351611133639032763641647255667324719299222003223177} a^{2} - \frac{4644183531506566003172493155555915304987392253396882655130519321}{13949321056260329000405502589745549650602011844947385996058017186} a - \frac{1506517368334575565343048374981904054705667147450920064717286772}{6974660528130164500202751294872774825301005922473692998029008593}$

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

Class group and class number

$C_{30}$, which has order $30$ (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:  \( 131820690.41331159 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-31}) \), \(\Q(\sqrt{-155}) \), \(\Q(\sqrt{5}, \sqrt{-31})\), 5.1.50000.1, 10.2.12500000000.1, 10.0.71572877500000000.2, 10.0.357864387500000000.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$31$31.10.5.2$x^{10} - 923521 x^{2} + 286291510$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
31.10.5.2$x^{10} - 923521 x^{2} + 286291510$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$