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Properties

Label 20.10.7857480558...9483.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,13^{10}\cdot 97^{2}\cdot 347^{7}$
Root discriminant $44.13$
Ramified primes $13, 97, 347$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T796

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![154547, -308835, 2270690, -4250894, 2876572, -4418610, 1429417, 1589249, -541338, 118185, -148199, -44206, 48529, -4464, -1590, 628, -310, 76, 6, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 6*x^18 + 76*x^17 - 310*x^16 + 628*x^15 - 1590*x^14 - 4464*x^13 + 48529*x^12 - 44206*x^11 - 148199*x^10 + 118185*x^9 - 541338*x^8 + 1589249*x^7 + 1429417*x^6 - 4418610*x^5 + 2876572*x^4 - 4250894*x^3 + 2270690*x^2 - 308835*x + 154547)
 
gp: K = bnfinit(x^20 - 7*x^19 + 6*x^18 + 76*x^17 - 310*x^16 + 628*x^15 - 1590*x^14 - 4464*x^13 + 48529*x^12 - 44206*x^11 - 148199*x^10 + 118185*x^9 - 541338*x^8 + 1589249*x^7 + 1429417*x^6 - 4418610*x^5 + 2876572*x^4 - 4250894*x^3 + 2270690*x^2 - 308835*x + 154547, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 6 x^{18} + 76 x^{17} - 310 x^{16} + 628 x^{15} - 1590 x^{14} - 4464 x^{13} + 48529 x^{12} - 44206 x^{11} - 148199 x^{10} + 118185 x^{9} - 541338 x^{8} + 1589249 x^{7} + 1429417 x^{6} - 4418610 x^{5} + 2876572 x^{4} - 4250894 x^{3} + 2270690 x^{2} - 308835 x + 154547 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-785748055861331850537130300409483=-\,13^{10}\cdot 97^{2}\cdot 347^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.13$
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:  $13, 97, 347$
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}{6004582852680123731026137090673169864063407921474505532050653490313361} a^{19} - \frac{2501343738423677249254592347309577169775997927072561905330242791305969}{6004582852680123731026137090673169864063407921474505532050653490313361} a^{18} - \frac{237536860740661452347506068190577361169132823928188153121353803281158}{545871168425465793729648826424833624005764356497682321095513953664851} a^{17} - \frac{2822053371425376978473295395141350046310286405044294502931374505150683}{6004582852680123731026137090673169864063407921474505532050653490313361} a^{16} + \frac{2944473347092019246005037871559186829224518334466632905147134691797903}{6004582852680123731026137090673169864063407921474505532050653490313361} a^{15} - \frac{379325002342789453796995429366942550154762123483130950605929502280366}{6004582852680123731026137090673169864063407921474505532050653490313361} a^{14} + \frac{2828646644343656668035803288098206024382786467629781581734887232408576}{6004582852680123731026137090673169864063407921474505532050653490313361} a^{13} + \frac{2342591366309421367622327075776371714903304334649446638414485542026375}{6004582852680123731026137090673169864063407921474505532050653490313361} a^{12} - \frac{2137260955479087853807420430613658674404028356455881774285101600773005}{6004582852680123731026137090673169864063407921474505532050653490313361} a^{11} + \frac{1377812356505086244340183134924615190557579114294056376079549502524358}{6004582852680123731026137090673169864063407921474505532050653490313361} a^{10} + \frac{2898555052532168940860230458898372250309060946465834151210943683217507}{6004582852680123731026137090673169864063407921474505532050653490313361} a^{9} - \frac{36814011471328361103074277228161491616029097442869196293596204568513}{545871168425465793729648826424833624005764356497682321095513953664851} a^{8} + \frac{1750475472876997890663968018906708462094634148581550627271413146896629}{6004582852680123731026137090673169864063407921474505532050653490313361} a^{7} - \frac{1716251406859323461030155555723727570719706272152448775019513002327661}{6004582852680123731026137090673169864063407921474505532050653490313361} a^{6} + \frac{2074151371593936222490543701546506170670140001367486498766889769665737}{6004582852680123731026137090673169864063407921474505532050653490313361} a^{5} - \frac{2013786752088856573880138221497005552303442481621172363091518644510255}{6004582852680123731026137090673169864063407921474505532050653490313361} a^{4} + \frac{264644790201350057337079564063040531029558502876168130829411192666392}{545871168425465793729648826424833624005764356497682321095513953664851} a^{3} + \frac{2194498158119760875066510875013589619403354157379681650048495988777472}{6004582852680123731026137090673169864063407921474505532050653490313361} a^{2} - \frac{170604341176559923767974126449388467646357737704901608497017092090938}{6004582852680123731026137090673169864063407921474505532050653490313361} a + \frac{1888445935900620305220188737146541381335284301134737378932548195912596}{6004582852680123731026137090673169864063407921474505532050653490313361}$

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:  $14$
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:  \( 1116604586.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T796:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n796 are not computed
Character table for t20n796 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.3.4511.1, 10.6.44707018837.1

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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
347Data not computed