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Properties

Label 20.0.10975122531...0000.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 41^{18}$
Root discriminant $126.48$
Ramified primes $2, 5, 41$
Class number $2182840$ (GRH)
Class group $[22, 22, 4510]$ (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![940204701, -42959304, 727173193, -73298264, 261654120, -30342304, 58673502, -6208538, 9013852, -807582, 1031934, -80862, 94062, -8202, 6781, -584, 390, -28, 15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 15*x^18 - 28*x^17 + 390*x^16 - 584*x^15 + 6781*x^14 - 8202*x^13 + 94062*x^12 - 80862*x^11 + 1031934*x^10 - 807582*x^9 + 9013852*x^8 - 6208538*x^7 + 58673502*x^6 - 30342304*x^5 + 261654120*x^4 - 73298264*x^3 + 727173193*x^2 - 42959304*x + 940204701)
 
gp: K = bnfinit(x^20 - 2*x^19 + 15*x^18 - 28*x^17 + 390*x^16 - 584*x^15 + 6781*x^14 - 8202*x^13 + 94062*x^12 - 80862*x^11 + 1031934*x^10 - 807582*x^9 + 9013852*x^8 - 6208538*x^7 + 58673502*x^6 - 30342304*x^5 + 261654120*x^4 - 73298264*x^3 + 727173193*x^2 - 42959304*x + 940204701, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 15 x^{18} - 28 x^{17} + 390 x^{16} - 584 x^{15} + 6781 x^{14} - 8202 x^{13} + 94062 x^{12} - 80862 x^{11} + 1031934 x^{10} - 807582 x^{9} + 9013852 x^{8} - 6208538 x^{7} + 58673502 x^{6} - 30342304 x^{5} + 261654120 x^{4} - 73298264 x^{3} + 727173193 x^{2} - 42959304 x + 940204701 \)

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:  \(1097512253107525471809057479895040000000000=2^{20}\cdot 5^{10}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $126.48$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(820=2^{2}\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{820}(1,·)$, $\chi_{820}(619,·)$, $\chi_{820}(441,·)$, $\chi_{820}(201,·)$, $\chi_{820}(139,·)$, $\chi_{820}(461,·)$, $\chi_{820}(141,·)$, $\chi_{820}(81,·)$, $\chi_{820}(599,·)$, $\chi_{820}(221,·)$, $\chi_{820}(739,·)$, $\chi_{820}(359,·)$, $\chi_{820}(681,·)$, $\chi_{820}(679,·)$, $\chi_{820}(59,·)$, $\chi_{820}(819,·)$, $\chi_{820}(119,·)$, $\chi_{820}(761,·)$, $\chi_{820}(379,·)$, $\chi_{820}(701,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{15} - \frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{3} a^{7} + \frac{2}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{3} a^{4} - \frac{2}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{15} - \frac{1}{9} a^{14} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{8} + \frac{2}{9} a^{7} + \frac{1}{9} a^{5} - \frac{2}{9} a^{4} - \frac{1}{9} a^{2}$, $\frac{1}{2241} a^{18} - \frac{40}{2241} a^{17} + \frac{10}{249} a^{16} + \frac{319}{2241} a^{15} - \frac{101}{2241} a^{14} + \frac{70}{747} a^{13} + \frac{77}{2241} a^{12} + \frac{158}{2241} a^{11} - \frac{50}{2241} a^{10} + \frac{9}{83} a^{9} + \frac{346}{2241} a^{8} - \frac{818}{2241} a^{7} + \frac{260}{747} a^{6} - \frac{223}{2241} a^{5} - \frac{478}{2241} a^{4} + \frac{67}{249} a^{3} - \frac{82}{2241} a^{2} - \frac{10}{249} a + \frac{107}{249}$, $\frac{1}{102812957539345748978196468243171643330537703996292256820827161} a^{19} + \frac{5803200741132395042059229981242932045094510655695417162424}{34270985846448582992732156081057214443512567998764085606942387} a^{18} + \frac{4050344498223487185803977460357817631534317378684623564505047}{102812957539345748978196468243171643330537703996292256820827161} a^{17} - \frac{5111372620617563486253777825772636495860025149237081412335839}{102812957539345748978196468243171643330537703996292256820827161} a^{16} - \frac{12995226219942553861402078377035719829694411473279380279951024}{102812957539345748978196468243171643330537703996292256820827161} a^{15} - \frac{1432861233308767899855392304470701882070758244778138890132744}{102812957539345748978196468243171643330537703996292256820827161} a^{14} + \frac{12790472631013076220827885735138595513693960672205352027541674}{102812957539345748978196468243171643330537703996292256820827161} a^{13} + \frac{2449904908470565431453489347202660972609724776376523392194277}{102812957539345748978196468243171643330537703996292256820827161} a^{12} + \frac{2358838039878017477378743413617879166769935194274789007586861}{34270985846448582992732156081057214443512567998764085606942387} a^{11} + \frac{11406761439409315851228832609722048226597336504356347963318827}{102812957539345748978196468243171643330537703996292256820827161} a^{10} - \frac{675776454124401970260469250607431179540326142127050189596374}{102812957539345748978196468243171643330537703996292256820827161} a^{9} - \frac{6502289771396727520383911920873920471064769443619502943242985}{102812957539345748978196468243171643330537703996292256820827161} a^{8} - \frac{24666868432142204565562624575216626409050787487048142871290789}{102812957539345748978196468243171643330537703996292256820827161} a^{7} - \frac{31011082228142134049910320892469735806140369637814678990926640}{102812957539345748978196468243171643330537703996292256820827161} a^{6} + \frac{194996762140607976205556293385399658595931050239557797450409}{1238710331799346373231282749917730642536598843328822371335267} a^{5} + \frac{9695208142117424807873731244781938321710233072480135334753412}{102812957539345748978196468243171643330537703996292256820827161} a^{4} + \frac{25516009618598477508926066560407269537096701441940997471109127}{102812957539345748978196468243171643330537703996292256820827161} a^{3} + \frac{30801870378630106335378177766242624502486176783778882081511864}{102812957539345748978196468243171643330537703996292256820827161} a^{2} + \frac{536305959980592599503629854666039462941149771877402441739537}{3807887316272064776970239564561912715945840888751565067438043} a - \frac{19685562205702218135094323806464666133731906586562717303532}{70083815636909167674298887691323546919248605314445982836283}$

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

Class group and class number

$C_{22}\times C_{22}\times C_{4510}$, which has order $2182840$ (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:  \( 5104264.636551031 \) (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

\(\Q(\sqrt{-205}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{-5}, \sqrt{41})\), 5.5.2825761.1, 10.0.1047622190060675200000.1, 10.0.25551760733187200000.2, 10.10.327381934393961.1

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 R ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$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}$
$41$41.10.9.1$x^{10} - 41$$10$$1$$9$$C_{10}$$[\ ]_{10}$
41.10.9.1$x^{10} - 41$$10$$1$$9$$C_{10}$$[\ ]_{10}$