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Properties

Label 20.8.32809060263...8125.4
Degree $20$
Signature $[8, 6]$
Discriminant $5^{15}\cdot 401^{10}$
Root discriminant $66.96$
Ramified primes $5, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14211, -312951, 448726, 1298090, -3408590, 3488487, -2143907, 1473553, -1052756, 438711, -144458, 50238, -20445, -1048, 1144, 695, 105, -19, -25, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 25*x^18 - 19*x^17 + 105*x^16 + 695*x^15 + 1144*x^14 - 1048*x^13 - 20445*x^12 + 50238*x^11 - 144458*x^10 + 438711*x^9 - 1052756*x^8 + 1473553*x^7 - 2143907*x^6 + 3488487*x^5 - 3408590*x^4 + 1298090*x^3 + 448726*x^2 - 312951*x + 14211)
 
gp: K = bnfinit(x^20 - x^19 - 25*x^18 - 19*x^17 + 105*x^16 + 695*x^15 + 1144*x^14 - 1048*x^13 - 20445*x^12 + 50238*x^11 - 144458*x^10 + 438711*x^9 - 1052756*x^8 + 1473553*x^7 - 2143907*x^6 + 3488487*x^5 - 3408590*x^4 + 1298090*x^3 + 448726*x^2 - 312951*x + 14211, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 25 x^{18} - 19 x^{17} + 105 x^{16} + 695 x^{15} + 1144 x^{14} - 1048 x^{13} - 20445 x^{12} + 50238 x^{11} - 144458 x^{10} + 438711 x^{9} - 1052756 x^{8} + 1473553 x^{7} - 2143907 x^{6} + 3488487 x^{5} - 3408590 x^{4} + 1298090 x^{3} + 448726 x^{2} - 312951 x + 14211 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3280906026328914297094848663330078125=5^{15}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.96$
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:  $5, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{57} a^{18} - \frac{1}{57} a^{17} + \frac{16}{57} a^{16} - \frac{1}{19} a^{15} + \frac{20}{57} a^{14} + \frac{2}{57} a^{13} + \frac{26}{57} a^{12} + \frac{1}{19} a^{11} + \frac{1}{57} a^{10} - \frac{9}{19} a^{9} + \frac{7}{19} a^{8} + \frac{5}{19} a^{7} - \frac{17}{57} a^{6} - \frac{23}{57} a^{5} + \frac{7}{19} a^{4} - \frac{1}{57} a^{3} + \frac{16}{57} a^{2} - \frac{4}{19} a - \frac{2}{19}$, $\frac{1}{759099987149377425283108441763173548593992409589189300079164170349} a^{19} + \frac{383735082300109751846313634645380721946546684316737839957905289}{44652940420551613251947555397833738152587788799364076475244951197} a^{18} - \frac{83542100677074852350972121715327040121975732990650048532833698555}{759099987149377425283108441763173548593992409589189300079164170349} a^{17} + \frac{55587078909970040632640610252166448491926999489244907983914403547}{759099987149377425283108441763173548593992409589189300079164170349} a^{16} - \frac{117538901280799035403449800909746109208436159176220821492469615771}{253033329049792475094369480587724516197997469863063100026388056783} a^{15} + \frac{108742685726626057195849913192266071936823428185690098234680746291}{759099987149377425283108441763173548593992409589189300079164170349} a^{14} - \frac{289805802193075566480487650794068574530669809590601376309647989831}{759099987149377425283108441763173548593992409589189300079164170349} a^{13} - \frac{339792542936965333034831130460948304117355979580950376442767817067}{759099987149377425283108441763173548593992409589189300079164170349} a^{12} + \frac{39134660454729224860668501556408071329770798036260694160933659284}{84344443016597491698123160195908172065999156621021033342129352261} a^{11} - \frac{6353656392648933860788050500635214817686598814905868949458266741}{84344443016597491698123160195908172065999156621021033342129352261} a^{10} + \frac{286324513115907899496602637327998313925473023943263360371532120839}{759099987149377425283108441763173548593992409589189300079164170349} a^{9} + \frac{1854533385453276439025266693111480683128157497171189843084613961}{13317543634199603952335235820406553484105129992792794738230950357} a^{8} - \frac{157942032598395273190716009892719448489760690038233737755006650584}{759099987149377425283108441763173548593992409589189300079164170349} a^{7} - \frac{7228261161358912015214993594607888099290491522884533191120995845}{39952630902598811857005707461219660452315389978378384214692851071} a^{6} + \frac{203254353502105946365289874471980844211796144357902736231158107274}{759099987149377425283108441763173548593992409589189300079164170349} a^{5} - \frac{27296703205867204172551570757125491899770518366206635174827754848}{84344443016597491698123160195908172065999156621021033342129352261} a^{4} + \frac{20298685193594702783024229311708072329052131742296400857182602092}{58392306703798263483316033981782580661076339199168407698397243873} a^{3} - \frac{329654259549413157466453038981640113938905772005582645152386974441}{759099987149377425283108441763173548593992409589189300079164170349} a^{2} - \frac{196951739187393153506012934450131407720682799701156462515791117110}{759099987149377425283108441763173548593992409589189300079164170349} a + \frac{123890417338494883313861097280590840788369537782930074027236465578}{253033329049792475094369480587724516197997469863063100026388056783}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 62045702529.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.160801.1, 10.10.80803005003125.1

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 $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
401Data not computed