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Properties

Label 20.0.16975624485...0800.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{2}\cdot 36497^{5}$
Root discriminant $32.47$
Ramified primes $2, 5, 36497$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1297, -10388, 46159, -120740, 191409, -182618, 96689, -13454, -20080, 17494, -7516, 2502, -853, 174, 78, -114, 70, -24, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 5*x^18 - 24*x^17 + 70*x^16 - 114*x^15 + 78*x^14 + 174*x^13 - 853*x^12 + 2502*x^11 - 7516*x^10 + 17494*x^9 - 20080*x^8 - 13454*x^7 + 96689*x^6 - 182618*x^5 + 191409*x^4 - 120740*x^3 + 46159*x^2 - 10388*x + 1297)
 
gp: K = bnfinit(x^20 - 2*x^19 + 5*x^18 - 24*x^17 + 70*x^16 - 114*x^15 + 78*x^14 + 174*x^13 - 853*x^12 + 2502*x^11 - 7516*x^10 + 17494*x^9 - 20080*x^8 - 13454*x^7 + 96689*x^6 - 182618*x^5 + 191409*x^4 - 120740*x^3 + 46159*x^2 - 10388*x + 1297, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 5 x^{18} - 24 x^{17} + 70 x^{16} - 114 x^{15} + 78 x^{14} + 174 x^{13} - 853 x^{12} + 2502 x^{11} - 7516 x^{10} + 17494 x^{9} - 20080 x^{8} - 13454 x^{7} + 96689 x^{6} - 182618 x^{5} + 191409 x^{4} - 120740 x^{3} + 46159 x^{2} - 10388 x + 1297 \)

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:  \(1697562448531136375219997900800=2^{20}\cdot 5^{2}\cdot 36497^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.47$
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:  $2, 5, 36497$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3078865357451524208298983284371091693344809} a^{19} + \frac{1315363855144836713273142748676679971240891}{3078865357451524208298983284371091693344809} a^{18} + \frac{469731947114834915563568309492188043287926}{3078865357451524208298983284371091693344809} a^{17} - \frac{83856685034357299621525334345881318216416}{3078865357451524208298983284371091693344809} a^{16} + \frac{1006846194044663108053495696779123677249894}{3078865357451524208298983284371091693344809} a^{15} - \frac{1199720546026324109958794735959139178860242}{3078865357451524208298983284371091693344809} a^{14} + \frac{1520836895487833717409672781899497356717668}{3078865357451524208298983284371091693344809} a^{13} - \frac{1223502640531025760064392091391934888310400}{3078865357451524208298983284371091693344809} a^{12} + \frac{646245838317462270558881058749357348666858}{3078865357451524208298983284371091693344809} a^{11} - \frac{1226698877338456644372480092314156592196672}{3078865357451524208298983284371091693344809} a^{10} - \frac{636523759856630913884637184294339145120355}{3078865357451524208298983284371091693344809} a^{9} - \frac{1389725285228058886493018745707269803774687}{3078865357451524208298983284371091693344809} a^{8} + \frac{769827826353034546943522926294293498785734}{3078865357451524208298983284371091693344809} a^{7} - \frac{808695379482920829986582964351903709692428}{3078865357451524208298983284371091693344809} a^{6} - \frac{917814615448521417181074841972730970551467}{3078865357451524208298983284371091693344809} a^{5} + \frac{1361211976233616322600032610160746256990023}{3078865357451524208298983284371091693344809} a^{4} + \frac{301884857507769628932759778452963801032147}{3078865357451524208298983284371091693344809} a^{3} + \frac{965283542800855591402857749800402565296915}{3078865357451524208298983284371091693344809} a^{2} + \frac{429147349438325676232017139385808159892111}{3078865357451524208298983284371091693344809} a + \frac{1458961645392241234017703331639251647939913}{3078865357451524208298983284371091693344809}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( \frac{17618731022959867959248}{4465877815618430433407869} a^{19} - \frac{10101074623062369874615}{4465877815618430433407869} a^{18} + \frac{72292419133595428682988}{4465877815618430433407869} a^{17} - \frac{321993102129212726949666}{4465877815618430433407869} a^{16} + \frac{768702395775617619904762}{4465877815618430433407869} a^{15} - \frac{898471964100147183743353}{4465877815618430433407869} a^{14} + \frac{82922137134440470643033}{4465877815618430433407869} a^{13} + \frac{3143477983963463152701297}{4465877815618430433407869} a^{12} - \frac{10446199950048833722389076}{4465877815618430433407869} a^{11} + \frac{28973715772546859965922560}{4465877815618430433407869} a^{10} - \frac{90806191763836050499291836}{4465877815618430433407869} a^{9} + \frac{177955232403748384259505989}{4465877815618430433407869} a^{8} - \frac{97039317701853641789631739}{4465877815618430433407869} a^{7} - \frac{375764471851052557227144968}{4465877815618430433407869} a^{6} + \frac{1150479769580264703486961464}{4465877815618430433407869} a^{5} - \frac{1541806340084264902664908350}{4465877815618430433407869} a^{4} + \frac{1153129618401089039402914133}{4465877815618430433407869} a^{3} - \frac{532300554970919881127707912}{4465877815618430433407869} a^{2} + \frac{147307136450668009056102309}{4465877815618430433407869} a - \frac{22804534311738093680440785}{4465877815618430433407869} \) (order $4$)
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:  \( 6876702.94829 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n797 are not computed
Character table for t20n797 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.5.36497.1, 10.0.1363999753216.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
36497Data not computed