Properties

Label 20.0.28753215211...8032.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 193^{5}$
Root discriminant $10.54$
Ramified primes $2, 193$
Class number $1$
Class group Trivial
Galois Group $D_5\wr C_2$ (as 20T48)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 6, -4, -2, 10, -10, -12, 43, -38, 21, -38, 43, -12, -10, 10, -2, -4, 6, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 6*x^18 - 4*x^17 - 2*x^16 + 10*x^15 - 10*x^14 - 12*x^13 + 43*x^12 - 38*x^11 + 21*x^10 - 38*x^9 + 43*x^8 - 12*x^7 - 10*x^6 + 10*x^5 - 2*x^4 - 4*x^3 + 6*x^2 - 4*x + 1)
gp: K = bnfinit(x^20 - 4*x^19 + 6*x^18 - 4*x^17 - 2*x^16 + 10*x^15 - 10*x^14 - 12*x^13 + 43*x^12 - 38*x^11 + 21*x^10 - 38*x^9 + 43*x^8 - 12*x^7 - 10*x^6 + 10*x^5 - 2*x^4 - 4*x^3 + 6*x^2 - 4*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 4 x^{19} \) \(\mathstrut +\mathstrut 6 x^{18} \) \(\mathstrut -\mathstrut 4 x^{17} \) \(\mathstrut -\mathstrut 2 x^{16} \) \(\mathstrut +\mathstrut 10 x^{15} \) \(\mathstrut -\mathstrut 10 x^{14} \) \(\mathstrut -\mathstrut 12 x^{13} \) \(\mathstrut +\mathstrut 43 x^{12} \) \(\mathstrut -\mathstrut 38 x^{11} \) \(\mathstrut +\mathstrut 21 x^{10} \) \(\mathstrut -\mathstrut 38 x^{9} \) \(\mathstrut +\mathstrut 43 x^{8} \) \(\mathstrut -\mathstrut 12 x^{7} \) \(\mathstrut -\mathstrut 10 x^{6} \) \(\mathstrut +\mathstrut 10 x^{5} \) \(\mathstrut -\mathstrut 2 x^{4} \) \(\mathstrut -\mathstrut 4 x^{3} \) \(\mathstrut +\mathstrut 6 x^{2} \) \(\mathstrut -\mathstrut 4 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(287532152115567788032=2^{30}\cdot 193^{5}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.54$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3092351} a^{18} + \frac{814921}{3092351} a^{17} - \frac{1435426}{3092351} a^{16} - \frac{88748}{3092351} a^{15} - \frac{715639}{3092351} a^{14} + \frac{636475}{3092351} a^{13} + \frac{4222}{181903} a^{12} + \frac{1063649}{3092351} a^{11} - \frac{1172759}{3092351} a^{10} + \frac{1031245}{3092351} a^{9} - \frac{1172759}{3092351} a^{8} + \frac{1063649}{3092351} a^{7} + \frac{4222}{181903} a^{6} + \frac{636475}{3092351} a^{5} - \frac{715639}{3092351} a^{4} - \frac{88748}{3092351} a^{3} - \frac{1435426}{3092351} a^{2} + \frac{814921}{3092351} a + \frac{1}{3092351}$, $\frac{1}{3092351} a^{19} + \frac{167338}{3092351} a^{17} - \frac{371927}{3092351} a^{16} + \frac{1080432}{3092351} a^{15} + \frac{318553}{3092351} a^{14} + \frac{169178}{3092351} a^{13} - \frac{349391}{3092351} a^{12} + \frac{1090514}{3092351} a^{11} + \frac{429979}{3092351} a^{10} - \frac{886942}{3092351} a^{9} + \frac{27199}{181903} a^{8} - \frac{757304}{3092351} a^{7} - \frac{776565}{3092351} a^{6} - \frac{618235}{3092351} a^{5} - \frac{406670}{3092351} a^{4} + \frac{360645}{3092351} a^{3} + \frac{531742}{3092351} a^{2} - \frac{1489586}{3092351} a - \frac{814921}{3092351}$

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:  \( \frac{13983957}{3092351} a^{19} - \frac{48093261}{3092351} a^{18} + \frac{56956252}{3092351} a^{17} - \frac{23728783}{3092351} a^{16} - \frac{41951534}{3092351} a^{15} + \frac{116454852}{3092351} a^{14} - \frac{74046976}{3092351} a^{13} - \frac{210439531}{3092351} a^{12} + \frac{484757959}{3092351} a^{11} - \frac{258982479}{3092351} a^{10} + \frac{143153370}{3092351} a^{9} - \frac{446034696}{3092351} a^{8} + \frac{356073431}{3092351} a^{7} + \frac{32021208}{3092351} a^{6} - \frac{128572265}{3092351} a^{5} + \frac{65436014}{3092351} a^{4} + \frac{15879883}{3092351} a^{3} - \frac{47652050}{3092351} a^{2} + \frac{55099029}{3092351} a - \frac{22996286}{3092351} \),  \( \frac{597373}{3092351} a^{19} - \frac{1725800}{3092351} a^{18} + \frac{1160595}{3092351} a^{17} + \frac{630385}{3092351} a^{16} - \frac{1980459}{3092351} a^{15} + \frac{2862794}{3092351} a^{14} + \frac{505371}{3092351} a^{13} - \frac{11246046}{3092351} a^{12} + \frac{13990972}{3092351} a^{11} + \frac{4739105}{3092351} a^{10} - \frac{1464155}{3092351} a^{9} - \frac{23934622}{3092351} a^{8} + \frac{10652414}{3092351} a^{7} + \frac{15175731}{3092351} a^{6} + \frac{1842283}{3092351} a^{5} - \frac{9047742}{3092351} a^{4} + \frac{1921838}{3092351} a^{3} - \frac{529797}{3092351} a^{2} - \frac{791977}{3092351} a + \frac{2920193}{3092351} \),  \( \frac{9912023}{3092351} a^{19} - \frac{33667745}{3092351} a^{18} + \frac{39480510}{3092351} a^{17} - \frac{16862401}{3092351} a^{16} - \frac{29098098}{3092351} a^{15} + \frac{81915764}{3092351} a^{14} - \frac{51522395}{3092351} a^{13} - \frac{147216621}{3092351} a^{12} + \frac{336631463}{3092351} a^{11} - \frac{180161191}{3092351} a^{10} + \frac{110583846}{3092351} a^{9} - \frac{310975607}{3092351} a^{8} + \frac{234681893}{3092351} a^{7} + \frac{16478816}{3092351} a^{6} - \frac{85491782}{3092351} a^{5} + \frac{56685460}{3092351} a^{4} + \frac{8151822}{3092351} a^{3} - \frac{33901382}{3092351} a^{2} + \frac{37567205}{3092351} a - \frac{16223477}{3092351} \),  \( \frac{813378}{181903} a^{19} - \frac{46372892}{3092351} a^{18} + \frac{53607127}{3092351} a^{17} - \frac{22785245}{3092351} a^{16} - \frac{39946474}{3092351} a^{15} + \frac{111747923}{3092351} a^{14} - \frac{68368494}{3092351} a^{13} - \frac{12050754}{181903} a^{12} + \frac{459533158}{3092351} a^{11} - \frac{239008636}{3092351} a^{10} + \frac{158302973}{3092351} a^{9} - \frac{433020439}{3092351} a^{8} + \frac{316227799}{3092351} a^{7} + \frac{1306111}{181903} a^{6} - \frac{107813883}{3092351} a^{5} + \frac{69682341}{3092351} a^{4} + \frac{10870453}{3092351} a^{3} - \frac{48601309}{3092351} a^{2} + \frac{56242178}{3092351} a - \frac{23447967}{3092351} \),  \( \frac{420587}{181903} a^{19} - \frac{22758229}{3092351} a^{18} + \frac{23229970}{3092351} a^{17} - \frac{6142781}{3092351} a^{16} - \frac{22609901}{3092351} a^{15} + \frac{52945981}{3092351} a^{14} - \frac{23659533}{3092351} a^{13} - \frac{6690265}{181903} a^{12} + \frac{217389973}{3092351} a^{11} - \frac{74246345}{3092351} a^{10} + \frac{54526109}{3092351} a^{9} - \frac{219881299}{3092351} a^{8} + \frac{130275955}{3092351} a^{7} + \frac{2893526}{181903} a^{6} - \frac{48012101}{3092351} a^{5} + \frac{14990676}{3092351} a^{4} + \frac{10084992}{3092351} a^{3} - \frac{21073303}{3092351} a^{2} + \frac{21278030}{3092351} a - \frac{5731913}{3092351} \),  \( \frac{9471583}{3092351} a^{19} - \frac{31844330}{3092351} a^{18} + \frac{36713517}{3092351} a^{17} - \frac{15488030}{3092351} a^{16} - \frac{26748782}{3092351} a^{15} + \frac{75600407}{3092351} a^{14} - \frac{46074492}{3092351} a^{13} - \frac{140613504}{3092351} a^{12} + \frac{313427917}{3092351} a^{11} - \frac{160494017}{3092351} a^{10} + \frac{6351541}{181903} a^{9} - \frac{307588083}{3092351} a^{8} + \frac{222594438}{3092351} a^{7} + \frac{16594249}{3092351} a^{6} - \frac{66657405}{3092351} a^{5} + \frac{40099327}{3092351} a^{4} + \frac{8755249}{3092351} a^{3} - \frac{28004161}{3092351} a^{2} + \frac{34124878}{3092351} a - \frac{16683041}{3092351} \),  \( \frac{6203593}{3092351} a^{19} - \frac{21648320}{3092351} a^{18} + \frac{25684762}{3092351} a^{17} - \frac{600795}{181903} a^{16} - \frac{1134072}{181903} a^{15} + \frac{53071820}{3092351} a^{14} - \frac{33855338}{3092351} a^{13} - \frac{94782746}{3092351} a^{12} + \frac{219833437}{3092351} a^{11} - \frac{116731915}{3092351} a^{10} + \frac{57031601}{3092351} a^{9} - \frac{200366926}{3092351} a^{8} + \frac{157314667}{3092351} a^{7} + \frac{20926317}{3092351} a^{6} - \frac{65589521}{3092351} a^{5} + \frac{33444951}{3092351} a^{4} + \frac{5030714}{3092351} a^{3} - \frac{1244080}{181903} a^{2} + \frac{25510300}{3092351} a - \frac{13320600}{3092351} \),  \( \frac{492432}{3092351} a^{19} - \frac{3851597}{3092351} a^{18} + \frac{8249188}{3092351} a^{17} - \frac{6267325}{3092351} a^{16} - \frac{463208}{3092351} a^{15} + \frac{10702160}{3092351} a^{14} - \frac{17236230}{3092351} a^{13} - \frac{2889407}{3092351} a^{12} + \frac{50562106}{3092351} a^{11} - \frac{64852990}{3092351} a^{10} + \frac{16571875}{3092351} a^{9} - \frac{36786463}{3092351} a^{8} + \frac{71776937}{3092351} a^{7} - \frac{15321555}{3092351} a^{6} - \frac{1228674}{181903} a^{5} + \frac{10937410}{3092351} a^{4} - \frac{10620625}{3092351} a^{3} - \frac{43217}{3092351} a^{2} + \frac{8846072}{3092351} a - \frac{5732550}{3092351} \),  \( \frac{9491484}{3092351} a^{19} - \frac{31482324}{3092351} a^{18} + \frac{35543505}{3092351} a^{17} - \frac{14466181}{3092351} a^{16} - \frac{27062002}{3092351} a^{15} + \frac{4411156}{181903} a^{14} - \frac{43952811}{3092351} a^{13} - \frac{141553205}{3092351} a^{12} + \frac{307585789}{3092351} a^{11} - \frac{150758217}{3092351} a^{10} + \frac{107070660}{3092351} a^{9} - \frac{302373989}{3092351} a^{8} + \frac{205194980}{3092351} a^{7} + \frac{21522375}{3092351} a^{6} - \frac{62806969}{3092351} a^{5} + \frac{48104819}{3092351} a^{4} + \frac{4638423}{3092351} a^{3} - \frac{31271310}{3092351} a^{2} + \frac{35341406}{3092351} a - \frac{14682861}{3092351} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 52.0455573525 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_5\wr C_2$ (as 20T48):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 200
The 14 conjugacy class representatives for $D_5\wr C_2$
Character table for $D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.12352.2, 10.2.1220575232.1 x5

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 25 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$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}$
$193$193.2.1.2$x^{2} + 965$$2$$1$$1$$C_2$$[\ ]_{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.1.2$x^{2} + 965$$2$$1$$1$$C_2$$[\ ]_{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.1.2$x^{2} + 965$$2$$1$$1$$C_2$$[\ ]_{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.1.2$x^{2} + 965$$2$$1$$1$$C_2$$[\ ]_{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.1.2$x^{2} + 965$$2$$1$$1$$C_2$$[\ ]_{2}$