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Properties

Label 20.0.17598488843...1792.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 7^{15}\cdot 11^{13}$
Root discriminant $57.85$
Ramified primes $2, 7, 11$
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![3806977113521, 222113646576, -1559801944752, -157170563398, 306909341114, 38600167906, -37554595669, -5238733956, 3167235334, 458406564, -194166885, -27555888, 8847515, 1164480, -299188, -34000, 7277, 634, -117, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 117*x^18 + 634*x^17 + 7277*x^16 - 34000*x^15 - 299188*x^14 + 1164480*x^13 + 8847515*x^12 - 27555888*x^11 - 194166885*x^10 + 458406564*x^9 + 3167235334*x^8 - 5238733956*x^7 - 37554595669*x^6 + 38600167906*x^5 + 306909341114*x^4 - 157170563398*x^3 - 1559801944752*x^2 + 222113646576*x + 3806977113521)
 
gp: K = bnfinit(x^20 - 6*x^19 - 117*x^18 + 634*x^17 + 7277*x^16 - 34000*x^15 - 299188*x^14 + 1164480*x^13 + 8847515*x^12 - 27555888*x^11 - 194166885*x^10 + 458406564*x^9 + 3167235334*x^8 - 5238733956*x^7 - 37554595669*x^6 + 38600167906*x^5 + 306909341114*x^4 - 157170563398*x^3 - 1559801944752*x^2 + 222113646576*x + 3806977113521, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 117 x^{18} + 634 x^{17} + 7277 x^{16} - 34000 x^{15} - 299188 x^{14} + 1164480 x^{13} + 8847515 x^{12} - 27555888 x^{11} - 194166885 x^{10} + 458406564 x^{9} + 3167235334 x^{8} - 5238733956 x^{7} - 37554595669 x^{6} + 38600167906 x^{5} + 306909341114 x^{4} - 157170563398 x^{3} - 1559801944752 x^{2} + 222113646576 x + 3806977113521 \)

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:  \(175984888437863295253314247476641792=2^{30}\cdot 7^{15}\cdot 11^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.85$
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:  $2, 7, 11$
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}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231} a^{19} + \frac{448176285423238472737498766625883113708827548017829151970691697831108138771921660585203570547260}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231} a^{18} - \frac{321266424478952918423073808807472773270819378476007319605079245513699630404445637759060941185884}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231} a^{17} + \frac{52058601704261801872756632050588764159304243088036308010670367928863828933012492618076724746531}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231} a^{16} + \frac{1447692057573693137681712229331212073789626309853815173774011509478607538799174253643447904876735}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231} a^{15} - \frac{1245842267745814593507012248015097118546933107903302550406844734643558073480115715488062536759388}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231} a^{14} + \frac{326625270730304673095534109164068516634476158248206176889752497615690023513358351901393492720463}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231} a^{13} - \frac{103345762079643683459628476909471567215645760943865230513847293571469046462425547480331205562333}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231} a^{12} + \frac{475825734832380449800126871256107805499071274995481814395419546749508019130617691306734618951670}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231} a^{11} + \frac{2446490210579669525266616264320892493436341320250346428864802821164078841099457969781539432484}{6658482503899511924637578819849824894349440056814667778726353143059798619185014163197692021529} a^{10} - \frac{39676703601214737547631834740364592492602182778874881027401684537913466144679470056605671668332}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231} a^{9} + \frac{83023814362597835753111724636576036864913308267248358790955258676397957480629471388132772678604}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231} a^{8} + \frac{1451848821337715899115377178895759368117420878530721764799834922053183135669147963078893780873164}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231} a^{7} + \frac{1146246391705645457838250961770395914600813213253672660544933106656894119460608987740901392552403}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231} a^{6} - \frac{290436159833402690436513585637512656362495922478394677648979170474636552201213767291872482678854}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231} a^{5} - \frac{745793876917816194960458882955605670819132822006170151067653651931851116919226668801277030178869}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231} a^{4} + \frac{1037223897259377403086950010858908283915965703972836126437188338999095977361043563574390839037953}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231} a^{3} + \frac{836824370520035170279045180549592673476063150644098496434754424339088443496690592451305600323340}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231} a^{2} - \frac{487017604761034759209316798883098571127488482047817048078813158558147987125391351790951327552191}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231} a - \frac{853390281463068208711754308705935737847630641776312956568479098244934544469913100936456859954799}{2923073819211885734915897101914073128619404184941639154860869029803251593822221217643786797451231}$

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

\(\Q(\sqrt{-7}) \), 4.0.241472.1, 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 R $20$ $20$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{10}$ $20$

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
$2$2.10.15.5$x^{10} + 14 x^{8} + 40 x^{6} - 144 x^{4} - 432 x^{2} + 33632$$2$$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}$
7Data not computed
$11$11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$