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Properties

Label 20.0.36779002319...5625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 11^{16}\cdot 31^{10}$
Root discriminant $84.78$
Ramified primes $5, 11, 31$
Class number $387066$ (GRH)
Class group $[387066]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12117934841, -7849045071, 10684140682, -5362253828, 4065065252, -1655398135, 905756590, -309045225, 134662880, -39343938, 14321274, -3621581, 1126555, -246419, 66113, -12358, 2847, -432, 81, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 81*x^18 - 432*x^17 + 2847*x^16 - 12358*x^15 + 66113*x^14 - 246419*x^13 + 1126555*x^12 - 3621581*x^11 + 14321274*x^10 - 39343938*x^9 + 134662880*x^8 - 309045225*x^7 + 905756590*x^6 - 1655398135*x^5 + 4065065252*x^4 - 5362253828*x^3 + 10684140682*x^2 - 7849045071*x + 12117934841)
 
gp: K = bnfinit(x^20 - 8*x^19 + 81*x^18 - 432*x^17 + 2847*x^16 - 12358*x^15 + 66113*x^14 - 246419*x^13 + 1126555*x^12 - 3621581*x^11 + 14321274*x^10 - 39343938*x^9 + 134662880*x^8 - 309045225*x^7 + 905756590*x^6 - 1655398135*x^5 + 4065065252*x^4 - 5362253828*x^3 + 10684140682*x^2 - 7849045071*x + 12117934841, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 81 x^{18} - 432 x^{17} + 2847 x^{16} - 12358 x^{15} + 66113 x^{14} - 246419 x^{13} + 1126555 x^{12} - 3621581 x^{11} + 14321274 x^{10} - 39343938 x^{9} + 134662880 x^{8} - 309045225 x^{7} + 905756590 x^{6} - 1655398135 x^{5} + 4065065252 x^{4} - 5362253828 x^{3} + 10684140682 x^{2} - 7849045071 x + 12117934841 \)

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:  \(367790023196388722242846533993759765625=5^{10}\cdot 11^{16}\cdot 31^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.78$
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:  $5, 11, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1705=5\cdot 11\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{1705}(1024,·)$, $\chi_{1705}(1,·)$, $\chi_{1705}(774,·)$, $\chi_{1705}(1611,·)$, $\chi_{1705}(1549,·)$, $\chi_{1705}(526,·)$, $\chi_{1705}(1489,·)$, $\chi_{1705}(466,·)$, $\chi_{1705}(1301,·)$, $\chi_{1705}(1241,·)$, $\chi_{1705}(991,·)$, $\chi_{1705}(929,·)$, $\chi_{1705}(619,·)$, $\chi_{1705}(1644,·)$, $\chi_{1705}(621,·)$, $\chi_{1705}(559,·)$, $\chi_{1705}(1456,·)$, $\chi_{1705}(309,·)$, $\chi_{1705}(1334,·)$, $\chi_{1705}(311,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{36079} a^{18} + \frac{2108}{36079} a^{17} + \frac{6495}{36079} a^{16} - \frac{4305}{36079} a^{15} - \frac{7932}{36079} a^{14} - \frac{1174}{36079} a^{13} - \frac{4569}{36079} a^{12} - \frac{8806}{36079} a^{11} + \frac{9263}{36079} a^{10} - \frac{246}{36079} a^{9} - \frac{10340}{36079} a^{8} - \frac{4501}{36079} a^{7} - \frac{9528}{36079} a^{6} - \frac{335}{36079} a^{5} + \frac{14561}{36079} a^{4} - \frac{12860}{36079} a^{3} + \frac{14154}{36079} a^{2} + \frac{35}{109} a - \frac{12537}{36079}$, $\frac{1}{54743777367199368656192860267304745668892867244540942406742788951807309} a^{19} - \frac{332956237259127521116071497269685450777660908234086619763986577410}{54743777367199368656192860267304745668892867244540942406742788951807309} a^{18} + \frac{27261757917897697889471357216686923818347556341046403010657879620890985}{54743777367199368656192860267304745668892867244540942406742788951807309} a^{17} - \frac{11427416640488621715811842228602856072092515359918955938181503115392749}{54743777367199368656192860267304745668892867244540942406742788951807309} a^{16} - \frac{26898875932109455272262368902243293873770195355948637696608265217269297}{54743777367199368656192860267304745668892867244540942406742788951807309} a^{15} + \frac{4063496350391779623319538796417050315734364922093209142970733611125958}{54743777367199368656192860267304745668892867244540942406742788951807309} a^{14} - \frac{3895350368631139011032436458037139880376959969182614195339149311494873}{54743777367199368656192860267304745668892867244540942406742788951807309} a^{13} - \frac{1339186172567210000071357577281114795117921754760414942698941988517447}{54743777367199368656192860267304745668892867244540942406742788951807309} a^{12} - \frac{8742736112642604121460661458547894072756380899367220076891297261846457}{54743777367199368656192860267304745668892867244540942406742788951807309} a^{11} - \frac{24709474012229581282293183332468073768949464117739046289964145481645924}{54743777367199368656192860267304745668892867244540942406742788951807309} a^{10} - \frac{17779847903953127733973617102361357119887995205197875343793789364544494}{54743777367199368656192860267304745668892867244540942406742788951807309} a^{9} + \frac{22453430711794348367671741913777424994161707004276935117139420962561118}{54743777367199368656192860267304745668892867244540942406742788951807309} a^{8} - \frac{19800523736135669245421603796628565514672586321301832159921447526907890}{54743777367199368656192860267304745668892867244540942406742788951807309} a^{7} - \frac{15977513421234703825610671049762856861522423774752744940878680025479840}{54743777367199368656192860267304745668892867244540942406742788951807309} a^{6} - \frac{3165474841772203179848705032413896536318248894597471090637145106114968}{54743777367199368656192860267304745668892867244540942406742788951807309} a^{5} - \frac{20825646708298036232634861776638269134160701849577556024762844173478953}{54743777367199368656192860267304745668892867244540942406742788951807309} a^{4} + \frac{19295944594780365234267282684341629158620664633624149173571963532155973}{54743777367199368656192860267304745668892867244540942406742788951807309} a^{3} + \frac{21452770177795001310918786057135246894851876784426964503373463874411267}{54743777367199368656192860267304745668892867244540942406742788951807309} a^{2} + \frac{353524502494645265410363371709766321759322982148798027364904199740757}{1273111101562776015260299075983831294625415517314905637366111370972263} a + \frac{5827088126630020360721962750067292361118190858314406236482801587110735}{54743777367199368656192860267304745668892867244540942406742788951807309}$

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

Class group and class number

$C_{387066}$, which has order $387066$ (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:  \( 140644.599182 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-31}) \), \(\Q(\sqrt{-155}) \), \(\Q(\sqrt{5}, \sqrt{-31})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.0.6136912772340031.1, 10.0.19177852413562596875.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$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}$