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Properties

Label 20.0.82300358279...5056.5
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{18}\cdot 13^{10}$
Root discriminant $88.26$
Ramified primes $2, 11, 13$
Class number $318300$ (GRH)
Class group $[318300]$ (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![1023598489, 153275916, 992360396, -296136026, 445313172, -157947368, 100228517, -36359542, 16965405, -5184914, 2319998, -564192, 259166, -45860, 21586, -2688, 1294, -100, 49, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 49*x^18 - 100*x^17 + 1294*x^16 - 2688*x^15 + 21586*x^14 - 45860*x^13 + 259166*x^12 - 564192*x^11 + 2319998*x^10 - 5184914*x^9 + 16965405*x^8 - 36359542*x^7 + 100228517*x^6 - 157947368*x^5 + 445313172*x^4 - 296136026*x^3 + 992360396*x^2 + 153275916*x + 1023598489)
 
gp: K = bnfinit(x^20 - 2*x^19 + 49*x^18 - 100*x^17 + 1294*x^16 - 2688*x^15 + 21586*x^14 - 45860*x^13 + 259166*x^12 - 564192*x^11 + 2319998*x^10 - 5184914*x^9 + 16965405*x^8 - 36359542*x^7 + 100228517*x^6 - 157947368*x^5 + 445313172*x^4 - 296136026*x^3 + 992360396*x^2 + 153275916*x + 1023598489, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 49 x^{18} - 100 x^{17} + 1294 x^{16} - 2688 x^{15} + 21586 x^{14} - 45860 x^{13} + 259166 x^{12} - 564192 x^{11} + 2319998 x^{10} - 5184914 x^{9} + 16965405 x^{8} - 36359542 x^{7} + 100228517 x^{6} - 157947368 x^{5} + 445313172 x^{4} - 296136026 x^{3} + 992360396 x^{2} + 153275916 x + 1023598489 \)

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:  \(823003582791542802072855307752419885056=2^{30}\cdot 11^{18}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.26$
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:  $2, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1144=2^{3}\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1144}(1,·)$, $\chi_{1144}(1093,·)$, $\chi_{1144}(129,·)$, $\chi_{1144}(521,·)$, $\chi_{1144}(909,·)$, $\chi_{1144}(337,·)$, $\chi_{1144}(729,·)$, $\chi_{1144}(857,·)$, $\chi_{1144}(1117,·)$, $\chi_{1144}(545,·)$, $\chi_{1144}(805,·)$, $\chi_{1144}(233,·)$, $\chi_{1144}(157,·)$, $\chi_{1144}(625,·)$, $\chi_{1144}(573,·)$, $\chi_{1144}(285,·)$, $\chi_{1144}(885,·)$, $\chi_{1144}(313,·)$, $\chi_{1144}(701,·)$, $\chi_{1144}(53,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2683} a^{12} + \frac{137}{2683} a^{11} + \frac{21}{2683} a^{10} + \frac{468}{2683} a^{9} + \frac{270}{2683} a^{8} - \frac{241}{2683} a^{7} + \frac{1151}{2683} a^{6} - \frac{908}{2683} a^{5} + \frac{199}{2683} a^{4} + \frac{983}{2683} a^{3} - \frac{813}{2683} a^{2} + \frac{940}{2683} a - \frac{1039}{2683}$, $\frac{1}{2683} a^{13} + \frac{33}{2683} a^{11} + \frac{274}{2683} a^{10} + \frac{546}{2683} a^{9} + \frac{331}{2683} a^{8} - \frac{711}{2683} a^{7} - \frac{298}{2683} a^{6} + \frac{1177}{2683} a^{5} + \frac{550}{2683} a^{4} - \frac{1334}{2683} a^{3} - \frac{365}{2683} a^{2} - \frac{1035}{2683} a + \frac{144}{2683}$, $\frac{1}{2683} a^{14} + \frac{1119}{2683} a^{11} - \frac{147}{2683} a^{10} + \frac{985}{2683} a^{9} + \frac{1111}{2683} a^{8} - \frac{394}{2683} a^{7} + \frac{756}{2683} a^{6} + \frac{1001}{2683} a^{5} + \frac{148}{2683} a^{4} - \frac{608}{2683} a^{3} - \frac{1036}{2683} a^{2} + \frac{1320}{2683} a - \frac{592}{2683}$, $\frac{1}{2683} a^{15} - \frac{519}{2683} a^{11} - \frac{1050}{2683} a^{10} + \frac{604}{2683} a^{9} + \frac{655}{2683} a^{8} - \frac{548}{2683} a^{7} + \frac{872}{2683} a^{6} - \frac{657}{2683} a^{5} - \frac{600}{2683} a^{4} - \frac{983}{2683} a^{3} - \frac{1153}{2683} a^{2} - \frac{716}{2683} a + \frac{902}{2683}$, $\frac{1}{61709} a^{16} + \frac{7}{61709} a^{14} + \frac{8}{61709} a^{12} + \frac{751}{2683} a^{11} + \frac{696}{2683} a^{10} - \frac{497}{2683} a^{9} - \frac{14144}{61709} a^{8} - \frac{588}{2683} a^{7} + \frac{15586}{61709} a^{6} - \frac{229}{2683} a^{5} + \frac{21753}{61709} a^{4} + \frac{941}{2683} a^{3} + \frac{22374}{61709} a^{2} - \frac{418}{2683} a + \frac{27831}{61709}$, $\frac{1}{61709} a^{17} + \frac{7}{61709} a^{15} + \frac{8}{61709} a^{13} - \frac{237}{2683} a^{11} - \frac{170}{2683} a^{10} - \frac{14029}{61709} a^{9} + \frac{550}{2683} a^{8} - \frac{17833}{61709} a^{7} - \frac{704}{2683} a^{6} - \frac{30158}{61709} a^{5} - \frac{943}{2683} a^{4} + \frac{12990}{61709} a^{3} + \frac{1104}{2683} a^{2} + \frac{20678}{61709} a - \frac{464}{2683}$, $\frac{1}{1916866667} a^{18} - \frac{4215}{1916866667} a^{17} + \frac{13832}{1916866667} a^{16} + \frac{128344}{1916866667} a^{15} + \frac{300287}{1916866667} a^{14} + \frac{200236}{1916866667} a^{13} + \frac{184292}{1916866667} a^{12} + \frac{20754358}{83342029} a^{11} - \frac{41140605}{1916866667} a^{10} - \frac{700329444}{1916866667} a^{9} + \frac{96476806}{1916866667} a^{8} + \frac{134845023}{1916866667} a^{7} - \frac{437617502}{1916866667} a^{6} + \frac{795855276}{1916866667} a^{5} - \frac{602968981}{1916866667} a^{4} - \frac{356705751}{1916866667} a^{3} + \frac{387234109}{1916866667} a^{2} + \frac{600351192}{1916866667} a - \frac{403363349}{1916866667}$, $\frac{1}{9513400271281636263345917664532921271819300458448333} a^{19} - \frac{1795924349180783358943689676401544304370608}{9513400271281636263345917664532921271819300458448333} a^{18} - \frac{271266345579429413041967649366340597399875174}{413626098751375489710692072370996577035621759062971} a^{17} - \frac{13529871370900630957040530078224746316025942047}{9513400271281636263345917664532921271819300458448333} a^{16} - \frac{1656005957089904242506869786924144671604402262106}{9513400271281636263345917664532921271819300458448333} a^{15} + \frac{1469903732885387355663057743006106406151442896953}{9513400271281636263345917664532921271819300458448333} a^{14} - \frac{1389650964010705163163019849911728097277789090380}{9513400271281636263345917664532921271819300458448333} a^{13} - \frac{250178715098137836157900331052387526661004942627}{9513400271281636263345917664532921271819300458448333} a^{12} - \frac{1867931783674162935566145902781783195548113872390590}{9513400271281636263345917664532921271819300458448333} a^{11} + \frac{1291458686239300670018824544088936853400486897141030}{9513400271281636263345917664532921271819300458448333} a^{10} - \frac{739917221074456147201333045300984542422264865987860}{9513400271281636263345917664532921271819300458448333} a^{9} - \frac{2166041047310564276543101982468415858438618564757882}{9513400271281636263345917664532921271819300458448333} a^{8} - \frac{3958693675685516761375042477028289067618916830718605}{9513400271281636263345917664532921271819300458448333} a^{7} + \frac{1951767981011173179929819690161514689617244950812366}{9513400271281636263345917664532921271819300458448333} a^{6} - \frac{267467333800752262329256600634154770130699451746319}{9513400271281636263345917664532921271819300458448333} a^{5} + \frac{125887640968938490663637654536681768953848879687359}{413626098751375489710692072370996577035621759062971} a^{4} + \frac{3405927438975695591071535777516117629926475971613487}{9513400271281636263345917664532921271819300458448333} a^{3} + \frac{723468553095166736072909833052000199703242951331343}{9513400271281636263345917664532921271819300458448333} a^{2} + \frac{3842755665639583620789478992368456891642227126706666}{9513400271281636263345917664532921271819300458448333} a + \frac{305066174419993785563198546131478499732607493933084}{9513400271281636263345917664532921271819300458448333}$

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

Class group and class number

$C_{318300}$, which has order $318300$ (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:  \( 530208.250733 \) (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

\(\Q(\sqrt{-286}) \), \(\Q(\sqrt{-143}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{-143})\), \(\Q(\zeta_{11})^+\), 10.0.28688039019625283584.1, 10.0.875489472034463.1, 10.10.7024111812608.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
11Data not computed
13Data not computed