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Properties

Label 20.4.49641192728...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{24}\cdot 5^{10}\cdot 13^{6}\cdot 29^{4}\cdot 31^{6}$
Root discriminant $60.92$
Ramified primes $2, 5, 13, 29, 31$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![345220, 602780, -2411536, 3841380, -4591244, 3755096, -1983912, 448344, 157422, -139922, 330, 41448, -23992, 5506, 438, -814, 381, -141, 44, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 44*x^18 - 141*x^17 + 381*x^16 - 814*x^15 + 438*x^14 + 5506*x^13 - 23992*x^12 + 41448*x^11 + 330*x^10 - 139922*x^9 + 157422*x^8 + 448344*x^7 - 1983912*x^6 + 3755096*x^5 - 4591244*x^4 + 3841380*x^3 - 2411536*x^2 + 602780*x + 345220)
 
gp: K = bnfinit(x^20 - 9*x^19 + 44*x^18 - 141*x^17 + 381*x^16 - 814*x^15 + 438*x^14 + 5506*x^13 - 23992*x^12 + 41448*x^11 + 330*x^10 - 139922*x^9 + 157422*x^8 + 448344*x^7 - 1983912*x^6 + 3755096*x^5 - 4591244*x^4 + 3841380*x^3 - 2411536*x^2 + 602780*x + 345220, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 44 x^{18} - 141 x^{17} + 381 x^{16} - 814 x^{15} + 438 x^{14} + 5506 x^{13} - 23992 x^{12} + 41448 x^{11} + 330 x^{10} - 139922 x^{9} + 157422 x^{8} + 448344 x^{7} - 1983912 x^{6} + 3755096 x^{5} - 4591244 x^{4} + 3841380 x^{3} - 2411536 x^{2} + 602780 x + 345220 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(496411927280973952950108160000000000=2^{24}\cdot 5^{10}\cdot 13^{6}\cdot 29^{4}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.92$
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:  $2, 5, 13, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{9}$, $\frac{1}{1764084016770158800757258646733598395287467757993102248268242} a^{19} - \frac{150363216565423733790245286192443411621683743928080930139859}{882042008385079400378629323366799197643733878996551124134121} a^{18} - \frac{25629641692825465338182252074964289548737168459008672519585}{1764084016770158800757258646733598395287467757993102248268242} a^{17} + \frac{206200648544213955007988347897171475184222645083027365627443}{882042008385079400378629323366799197643733878996551124134121} a^{16} + \frac{96972076835034195495552264399330974790963529367798598405661}{1764084016770158800757258646733598395287467757993102248268242} a^{15} - \frac{131126446418175755193484940897103097609093526991185706625525}{1764084016770158800757258646733598395287467757993102248268242} a^{14} - \frac{177801277640024383743830526043150070004331908549055037793521}{1764084016770158800757258646733598395287467757993102248268242} a^{13} + \frac{182583903236728874969291478716447827511441274000946731727852}{882042008385079400378629323366799197643733878996551124134121} a^{12} + \frac{186863030344309827829756864488298505261040027066620556036030}{882042008385079400378629323366799197643733878996551124134121} a^{11} + \frac{50886253897682597646558650481337367776118043751821328318305}{882042008385079400378629323366799197643733878996551124134121} a^{10} + \frac{457149882540596411305945269847916293007035184726653268960119}{1764084016770158800757258646733598395287467757993102248268242} a^{9} + \frac{42061924415465178797949378229472186232618600964920297547585}{882042008385079400378629323366799197643733878996551124134121} a^{8} - \frac{354737338248800054293697026211829452034127191377420643604487}{882042008385079400378629323366799197643733878996551124134121} a^{7} + \frac{362800046240154469804238739974521011962387146104819287251902}{882042008385079400378629323366799197643733878996551124134121} a^{6} + \frac{303677486697427612678582626626077089287966964030933336583857}{882042008385079400378629323366799197643733878996551124134121} a^{5} - \frac{150504684107941057206924384465523174580051699887949941904586}{882042008385079400378629323366799197643733878996551124134121} a^{4} + \frac{417422791192068722411565812645239024634420258321877917310806}{882042008385079400378629323366799197643733878996551124134121} a^{3} + \frac{57424291448256782559405889564607464046750082592414082601011}{882042008385079400378629323366799197643733878996551124134121} a^{2} + \frac{420533746630019738177344404674414064939880233656413580285472}{882042008385079400378629323366799197643733878996551124134121} a - \frac{314517009352794121091672067246594247726679882820435491541695}{882042008385079400378629323366799197643733878996551124134121}$

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:  $11$
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:  \( 13722842007.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.109268775200000.1

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

Sibling fields

Degree 20 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
$13$13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.10.0.1$x^{10} + x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.6.5.4$x^{6} + 217$$6$$1$$5$$C_6$$[\ ]_{6}$
31.8.0.1$x^{8} - x + 22$$1$$8$$0$$C_8$$[\ ]^{8}$