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Properties

Label 20.0.15432571213...2437.1
Degree $20$
Signature $[0, 10]$
Discriminant $7^{15}\cdot 11^{9}\cdot 13^{10}$
Root discriminant $45.65$
Ramified primes $7, 11, 13$
Class number Not computed
Class group Not computed
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27778159711, 17038017814, 6342631671, -1144870468, -87670111, -143712065, -16898855, 19766269, 8140696, -2473867, -97899, -142599, 43163, 2870, 5758, -1600, 150, -40, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 40*x^17 + 150*x^16 - 1600*x^15 + 5758*x^14 + 2870*x^13 + 43163*x^12 - 142599*x^11 - 97899*x^10 - 2473867*x^9 + 8140696*x^8 + 19766269*x^7 - 16898855*x^6 - 143712065*x^5 - 87670111*x^4 - 1144870468*x^3 + 6342631671*x^2 + 17038017814*x + 27778159711)
 
gp: K = bnfinit(x^20 - 2*x^19 - 40*x^17 + 150*x^16 - 1600*x^15 + 5758*x^14 + 2870*x^13 + 43163*x^12 - 142599*x^11 - 97899*x^10 - 2473867*x^9 + 8140696*x^8 + 19766269*x^7 - 16898855*x^6 - 143712065*x^5 - 87670111*x^4 - 1144870468*x^3 + 6342631671*x^2 + 17038017814*x + 27778159711, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 40 x^{17} + 150 x^{16} - 1600 x^{15} + 5758 x^{14} + 2870 x^{13} + 43163 x^{12} - 142599 x^{11} - 97899 x^{10} - 2473867 x^{9} + 8140696 x^{8} + 19766269 x^{7} - 16898855 x^{6} - 143712065 x^{5} - 87670111 x^{4} - 1144870468 x^{3} + 6342631671 x^{2} + 17038017814 x + 27778159711 \)

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:  \(1543257121397609899589781498462437=7^{15}\cdot 11^{9}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.65$
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:  $7, 11, 13$
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}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a^{19} - \frac{9345563151015772925655619314989538738382194688235509635970931402168151133447561061681295526691206754}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a^{18} - \frac{14167335755558727470887444095771867297006097999866843332377181658096486261399529831363831201405289055}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a^{17} - \frac{5462849368265463875120069837091642864516985875227686609197782643944927081049018139654771339147010715}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a^{16} - \frac{16237408608017139952196548977441386494061029678449104777949821123523047389583376428338605661946968021}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a^{15} + \frac{13606064389400751056830536457846185608403193980265387014212101472920941793046713199696361749215182635}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a^{14} + \frac{9847179662634602637506670879278925826582830970154705121507176624268430308286039446907044910338465195}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a^{13} - \frac{17686867832413115812700131136591011488428964409630200737877413642094228906235388762921207552773466681}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a^{12} - \frac{7120201180249391239778980819377993050891420235631336134897583791539622844153386781950773293872618048}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a^{11} - \frac{140534887213268222084969473979625025971170594536181863864054155142678752905821915760272776563501895}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a^{10} + \frac{11414215593032479128811055691035125803322708438770976969427273564155764196054095420340263114102046068}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a^{9} + \frac{700717938357229818824325092673059589012821975289384224766588903428223770442589152504770836107222001}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a^{8} + \frac{21344303327357999571438963345470606080716407279275897457110227972181121065687964724014937418622884038}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a^{7} + \frac{16668090998570229878469187595314440279434072299428523983185764025167515288745543459723008108827646055}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a^{6} + \frac{19617699757345520109121958914893424820193593298682686091014370726595325620691688069083089887884829061}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a^{5} + \frac{23818067995931393166967016486612989949664390296504983135169654245673855620613195574554145821847701519}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a^{4} + \frac{11277571773679239303335517093159693608777923922324607442724125198376385711704066179395422937728472116}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a^{3} - \frac{520421995763103642301835422795253947067408093031767714247570622504091294930206256214422546243523291}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a^{2} - \frac{5277381238370055722271858493627803321085757852908135086884552739017915011773849328628494556620631497}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173} a + \frac{4329986371964665080048630023712542982395346982990847671487748701465298174916044367844138611501266899}{48346723203221054751051901074482820481019879225923915622680411176844662956288098957865280431794430173}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.637637.1, 10.0.246071287.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ $20$ $20$ R R R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ $20$

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
7Data not computed
$11$11.10.9.5$x^{10} - 8019$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
13Data not computed