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Properties

Label 20.4.18078415936...0000.3
Degree $20$
Signature $[4, 8]$
Discriminant $2^{28}\cdot 5^{22}\cdot 7^{10}$
Root discriminant $41.01$
Ramified primes $2, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![219389, 246960, -941220, 670680, 1225, -9656, 192780, 280, -4190, 16200, -25920, -240, 830, -2960, 0, -8, -115, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 115*x^16 - 8*x^15 - 2960*x^13 + 830*x^12 - 240*x^11 - 25920*x^10 + 16200*x^9 - 4190*x^8 + 280*x^7 + 192780*x^6 - 9656*x^5 + 1225*x^4 + 670680*x^3 - 941220*x^2 + 246960*x + 219389)
 
gp: K = bnfinit(x^20 - 115*x^16 - 8*x^15 - 2960*x^13 + 830*x^12 - 240*x^11 - 25920*x^10 + 16200*x^9 - 4190*x^8 + 280*x^7 + 192780*x^6 - 9656*x^5 + 1225*x^4 + 670680*x^3 - 941220*x^2 + 246960*x + 219389, 1)
 

Normalized defining polynomial

\( x^{20} - 115 x^{16} - 8 x^{15} - 2960 x^{13} + 830 x^{12} - 240 x^{11} - 25920 x^{10} + 16200 x^{9} - 4190 x^{8} + 280 x^{7} + 192780 x^{6} - 9656 x^{5} + 1225 x^{4} + 670680 x^{3} - 941220 x^{2} + 246960 x + 219389 \)

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:  \(180784159360000000000000000000000=2^{28}\cdot 5^{22}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.01$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{12} a^{16} - \frac{1}{12} a^{15} + \frac{1}{12} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{12} a^{10} + \frac{5}{12} a^{9} - \frac{1}{4} a^{8} - \frac{1}{12} a^{7} - \frac{5}{12} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{5}{12} a^{2} - \frac{1}{12} a - \frac{1}{3}$, $\frac{1}{12} a^{17} + \frac{1}{12} a^{14} - \frac{1}{4} a^{12} + \frac{1}{6} a^{11} + \frac{1}{12} a^{10} - \frac{1}{3} a^{9} + \frac{5}{12} a^{8} - \frac{1}{2} a^{7} + \frac{1}{12} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{3} a^{3} + \frac{1}{12} a^{2} + \frac{1}{12} a - \frac{1}{12}$, $\frac{1}{12} a^{18} + \frac{1}{12} a^{15} - \frac{1}{4} a^{13} + \frac{1}{6} a^{12} + \frac{1}{12} a^{11} + \frac{1}{6} a^{10} + \frac{5}{12} a^{9} + \frac{1}{12} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{3} a^{4} + \frac{1}{12} a^{3} - \frac{5}{12} a^{2} - \frac{1}{12} a - \frac{1}{2}$, $\frac{1}{377916175497653544303102536673529265995555910520153658784436} a^{19} + \frac{8282488178786516317593082329967895261227019082225385354343}{377916175497653544303102536673529265995555910520153658784436} a^{18} + \frac{721441974590188073530190997625540991639172718227369477495}{125972058499217848101034178891176421998518636840051219594812} a^{17} - \frac{1717219335019333795474189563801224465794932527622675348851}{377916175497653544303102536673529265995555910520153658784436} a^{16} - \frac{32781111310108964470960621120928945328773959345144133795137}{377916175497653544303102536673529265995555910520153658784436} a^{15} + \frac{5807255523110082294422108579021948446408172311862906177299}{62986029249608924050517089445588210999259318420025609797406} a^{14} - \frac{46280647401772276048374858224962706006214874920807922338739}{377916175497653544303102536673529265995555910520153658784436} a^{13} + \frac{34365453757700139858669538568429271328381596039145057131847}{188958087748826772151551268336764632997777955260076829392218} a^{12} - \frac{11717641871820888796960958990284033843385552701516914579863}{377916175497653544303102536673529265995555910520153658784436} a^{11} - \frac{311382315641089404554029883824668795445298125226903959834}{1657527085516024317118870774883900289454192590000673942037} a^{10} + \frac{165104680186864805376352592114800877100567497881349223709825}{377916175497653544303102536673529265995555910520153658784436} a^{9} - \frac{5484930871490514005507434521025683557644671072888085650209}{94479043874413386075775634168382316498888977630038414696109} a^{8} - \frac{181779525114968372624524916897629930254122048616573912357925}{377916175497653544303102536673529265995555910520153658784436} a^{7} - \frac{12319557990530134133414811186944329383060312361786693782331}{62986029249608924050517089445588210999259318420025609797406} a^{6} - \frac{2190673025430744382919772154367301981748370273899339460557}{377916175497653544303102536673529265995555910520153658784436} a^{5} + \frac{2919658348017678276883541263043605317186524728774245259279}{188958087748826772151551268336764632997777955260076829392218} a^{4} + \frac{14323266327343789804485306605201519115441863725803859274690}{31493014624804462025258544722794105499629659210012804898703} a^{3} - \frac{138592369241749922300622518113745870975588024730929567801171}{377916175497653544303102536673529265995555910520153658784436} a^{2} - \frac{38502185259308357221565223764784561507841634317431034211215}{188958087748826772151551268336764632997777955260076829392218} a - \frac{5695453162761998794620546734494203470646067426720591077863}{125972058499217848101034178891176421998518636840051219594812}$

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:  $11$
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:  \( 241063577.29739463 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{7}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{7})\), 5.1.50000.1, 10.2.2689120000000000.5, 10.2.13445600000000000.2, 10.2.12500000000.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
5Data not computed
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$