Properties

Label 20.0.69562352381...8025.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{2}\cdot 23^{2}\cdot 47^{10}$
Root discriminant $11.02$
Ramified primes $5, 23, 47$
Class number $1$
Class group Trivial
Galois Group 20T412

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 6, -9, 10, -22, 61, -115, 143, -122, 78, -51, 47, -44, 33, -23, 20, -18, 12, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 12*x^18 - 18*x^17 + 20*x^16 - 23*x^15 + 33*x^14 - 44*x^13 + 47*x^12 - 51*x^11 + 78*x^10 - 122*x^9 + 143*x^8 - 115*x^7 + 61*x^6 - 22*x^5 + 10*x^4 - 9*x^3 + 6*x^2 - 2*x + 1)
gp: K = bnfinit(x^20 - 5*x^19 + 12*x^18 - 18*x^17 + 20*x^16 - 23*x^15 + 33*x^14 - 44*x^13 + 47*x^12 - 51*x^11 + 78*x^10 - 122*x^9 + 143*x^8 - 115*x^7 + 61*x^6 - 22*x^5 + 10*x^4 - 9*x^3 + 6*x^2 - 2*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 5 x^{19} \) \(\mathstrut +\mathstrut 12 x^{18} \) \(\mathstrut -\mathstrut 18 x^{17} \) \(\mathstrut +\mathstrut 20 x^{16} \) \(\mathstrut -\mathstrut 23 x^{15} \) \(\mathstrut +\mathstrut 33 x^{14} \) \(\mathstrut -\mathstrut 44 x^{13} \) \(\mathstrut +\mathstrut 47 x^{12} \) \(\mathstrut -\mathstrut 51 x^{11} \) \(\mathstrut +\mathstrut 78 x^{10} \) \(\mathstrut -\mathstrut 122 x^{9} \) \(\mathstrut +\mathstrut 143 x^{8} \) \(\mathstrut -\mathstrut 115 x^{7} \) \(\mathstrut +\mathstrut 61 x^{6} \) \(\mathstrut -\mathstrut 22 x^{5} \) \(\mathstrut +\mathstrut 10 x^{4} \) \(\mathstrut -\mathstrut 9 x^{3} \) \(\mathstrut +\mathstrut 6 x^{2} \) \(\mathstrut -\mathstrut 2 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(695623523818852398025=5^{2}\cdot 23^{2}\cdot 47^{10}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.02$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 23, 47$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} - \frac{1}{5} a^{17} + \frac{2}{5} a^{16} + \frac{1}{5} a^{15} + \frac{2}{5} a^{14} - \frac{1}{5} a^{13} + \frac{2}{5} a^{12} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{26902105} a^{19} - \frac{826716}{26902105} a^{18} + \frac{1853142}{26902105} a^{17} - \frac{13322124}{26902105} a^{16} + \frac{2221077}{26902105} a^{15} + \frac{9008584}{26902105} a^{14} - \frac{8206148}{26902105} a^{13} - \frac{167017}{5380421} a^{12} + \frac{275226}{5380421} a^{11} + \frac{1309864}{26902105} a^{10} + \frac{11455339}{26902105} a^{9} - \frac{1793000}{5380421} a^{8} - \frac{6627726}{26902105} a^{7} + \frac{8459511}{26902105} a^{6} + \frac{13406381}{26902105} a^{5} + \frac{1900881}{26902105} a^{4} + \frac{1850773}{26902105} a^{3} - \frac{2556158}{26902105} a^{2} + \frac{545226}{26902105} a - \frac{2064651}{5380421}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( \frac{8629478}{26902105} a^{19} - \frac{9955291}{5380421} a^{18} + \frac{132997148}{26902105} a^{17} - \frac{214538581}{26902105} a^{16} + \frac{238933399}{26902105} a^{15} - \frac{248747342}{26902105} a^{14} + \frac{347576328}{26902105} a^{13} - \frac{505822539}{26902105} a^{12} + \frac{111980523}{5380421} a^{11} - \frac{552928958}{26902105} a^{10} + \frac{799541394}{26902105} a^{9} - \frac{1360398548}{26902105} a^{8} + \frac{1727453482}{26902105} a^{7} - \frac{279916378}{5380421} a^{6} + \frac{623425746}{26902105} a^{5} - \frac{93024999}{26902105} a^{4} + \frac{6233462}{26902105} a^{3} - \frac{19004765}{5380421} a^{2} + \frac{58204789}{26902105} a - \frac{9622107}{26902105} \),  \( \frac{720502}{26902105} a^{19} - \frac{11024627}{26902105} a^{18} + \frac{41046134}{26902105} a^{17} - \frac{80432773}{26902105} a^{16} + \frac{99411044}{26902105} a^{15} - \frac{102594707}{26902105} a^{14} + \frac{133101129}{26902105} a^{13} - \frac{40629816}{5380421} a^{12} + \frac{48510865}{5380421} a^{11} - \frac{235232722}{26902105} a^{10} + \frac{297867228}{26902105} a^{9} - \frac{101710215}{5380421} a^{8} + \frac{721568513}{26902105} a^{7} - \frac{707181638}{26902105} a^{6} + \frac{446348272}{26902105} a^{5} - \frac{185918023}{26902105} a^{4} + \frac{55911616}{26902105} a^{3} - \frac{79549331}{26902105} a^{2} + \frac{65690452}{26902105} a - \frac{6376722}{5380421} \),  \( \frac{3174338}{26902105} a^{19} - \frac{7953784}{26902105} a^{18} + \frac{9464802}{26902105} a^{17} - \frac{9847374}{26902105} a^{16} + \frac{4153904}{5380421} a^{15} - \frac{7822007}{5380421} a^{14} + \frac{37907162}{26902105} a^{13} - \frac{26991292}{26902105} a^{12} + \frac{9110092}{5380421} a^{11} - \frac{92082978}{26902105} a^{10} + \frac{112886708}{26902105} a^{9} - \frac{81057429}{26902105} a^{8} + \frac{94915652}{26902105} a^{7} - \frac{140287126}{26902105} a^{6} + \frac{154587697}{26902105} a^{5} - \frac{81838038}{26902105} a^{4} + \frac{11286638}{26902105} a^{3} - \frac{20312987}{26902105} a^{2} + \frac{27728581}{26902105} a - \frac{25008006}{26902105} \),  \( \frac{112974}{5380421} a^{19} + \frac{1314755}{5380421} a^{18} - \frac{6077644}{5380421} a^{17} + \frac{11529554}{5380421} a^{16} - \frac{13502021}{5380421} a^{15} + \frac{12995403}{5380421} a^{14} - \frac{18684589}{5380421} a^{13} + \frac{29691550}{5380421} a^{12} - \frac{32636701}{5380421} a^{11} + \frac{29758878}{5380421} a^{10} - \frac{40238312}{5380421} a^{9} + \frac{73864934}{5380421} a^{8} - \frac{102037079}{5380421} a^{7} + \frac{93602325}{5380421} a^{6} - \frac{59067774}{5380421} a^{5} + \frac{22908405}{5380421} a^{4} - \frac{10072421}{5380421} a^{3} + \frac{9322862}{5380421} a^{2} - \frac{9458326}{5380421} a + \frac{645590}{5380421} \),  \( \frac{1850671}{5380421} a^{19} - \frac{8191297}{5380421} a^{18} + \frac{16628251}{5380421} a^{17} - \frac{20125537}{5380421} a^{16} + \frac{18702560}{5380421} a^{15} - \frac{24904629}{5380421} a^{14} + \frac{41460659}{5380421} a^{13} - \frac{49268205}{5380421} a^{12} + \frac{43830879}{5380421} a^{11} - \frac{53645332}{5380421} a^{10} + \frac{101956164}{5380421} a^{9} - \frac{150895611}{5380421} a^{8} + \frac{139960079}{5380421} a^{7} - \frac{80496341}{5380421} a^{6} + \frac{27511404}{5380421} a^{5} - \frac{12984226}{5380421} a^{4} + \frac{17431767}{5380421} a^{3} - \frac{12208714}{5380421} a^{2} + \frac{5933569}{5380421} a + \frac{884483}{5380421} \),  \( \frac{255437}{5380421} a^{19} - \frac{10076999}{26902105} a^{18} + \frac{33293264}{26902105} a^{17} - \frac{58732853}{26902105} a^{16} + \frac{66049046}{26902105} a^{15} - \frac{68366358}{26902105} a^{14} + \frac{97279464}{26902105} a^{13} - \frac{147832683}{26902105} a^{12} + \frac{33136564}{5380421} a^{11} - \frac{31412264}{5380421} a^{10} + \frac{221683409}{26902105} a^{9} - \frac{379213211}{26902105} a^{8} + \frac{101390650}{5380421} a^{7} - \frac{444863456}{26902105} a^{6} + \frac{239643291}{26902105} a^{5} - \frac{105993739}{26902105} a^{4} + \frac{58340706}{26902105} a^{3} - \frac{73170112}{26902105} a^{2} + \frac{58045937}{26902105} a - \frac{6298359}{26902105} \),  \( \frac{110418}{5380421} a^{19} - \frac{5903431}{26902105} a^{18} + \frac{19494051}{26902105} a^{17} - \frac{40497212}{26902105} a^{16} + \frac{57976714}{26902105} a^{15} - \frac{68988472}{26902105} a^{14} + \frac{83536256}{26902105} a^{13} - \frac{112171422}{26902105} a^{12} + \frac{27954984}{5380421} a^{11} - \frac{30816275}{5380421} a^{10} + \frac{182841321}{26902105} a^{9} - \frac{274310624}{26902105} a^{8} + \frac{78419162}{5380421} a^{7} - \frac{434270109}{26902105} a^{6} + \frac{350888794}{26902105} a^{5} - \frac{214322021}{26902105} a^{4} + \frac{72839354}{26902105} a^{3} - \frac{14787393}{26902105} a^{2} - \frac{31493452}{26902105} a + \frac{1123509}{26902105} \),  \( \frac{5814592}{5380421} a^{19} - \frac{132604787}{26902105} a^{18} + \frac{295380812}{26902105} a^{17} - \frac{416023759}{26902105} a^{16} + \frac{445308733}{26902105} a^{15} - \frac{526859479}{26902105} a^{14} + \frac{765317402}{26902105} a^{13} - \frac{986295504}{26902105} a^{12} + \frac{205963862}{5380421} a^{11} - \frac{231514638}{5380421} a^{10} + \frac{1844620507}{26902105} a^{9} - \frac{2819921348}{26902105} a^{8} + \frac{630604173}{5380421} a^{7} - \frac{2357123378}{26902105} a^{6} + \frac{1058611698}{26902105} a^{5} - \frac{229742482}{26902105} a^{4} + \frac{81090953}{26902105} a^{3} - \frac{171774001}{26902105} a^{2} + \frac{134565491}{26902105} a - \frac{39965717}{26902105} \),  \( \frac{8117057}{26902105} a^{19} - \frac{39823612}{26902105} a^{18} + \frac{96959709}{26902105} a^{17} - \frac{146764758}{26902105} a^{16} + \frac{162944639}{26902105} a^{15} - \frac{175042152}{26902105} a^{14} + \frac{244445139}{26902105} a^{13} - \frac{67916335}{5380421} a^{12} + \frac{74330682}{5380421} a^{11} - \frac{385817322}{26902105} a^{10} + \frac{575902678}{26902105} a^{9} - \frac{188742944}{5380421} a^{8} + \frac{1146052408}{26902105} a^{7} - \frac{886501798}{26902105} a^{6} + \frac{375539937}{26902105} a^{5} - \frac{8385008}{26902105} a^{4} - \frac{31853774}{26902105} a^{3} - \frac{16488916}{26902105} a^{2} + \frac{45932647}{26902105} a - \frac{2945096}{5380421} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 86.800744447 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

20T412:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 10240
The 208 conjugacy class representatives for t20n412 are not computed
Character table for t20n412 is not computed

Intermediate fields

5.1.2209.1, 10.0.1146725035.1, 10.2.5274935161.1, 10.0.561163315.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$23$23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
47.8.6.1$x^{8} - 13489 x^{4} + 63091249$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$