Properties

Label 20.0.35754806217...3081.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 7^{10}\cdot 11^{8}$
Root discriminant $11.96$
Ramified primes $3, 7, 11$
Class number $1$
Class group Trivial
Galois Group $C_{10}\times D_5$ (as 20T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 0, 6, -10, 5, 1, -6, 20, -26, 10, 6, -1, -13, 18, -9, -2, 3, 2, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 2*x^18 + 3*x^17 - 2*x^16 - 9*x^15 + 18*x^14 - 13*x^13 - x^12 + 6*x^11 + 10*x^10 - 26*x^9 + 20*x^8 - 6*x^7 + x^6 + 5*x^5 - 10*x^4 + 6*x^3 - 2*x + 1)
gp: K = bnfinit(x^20 - 3*x^19 + 2*x^18 + 3*x^17 - 2*x^16 - 9*x^15 + 18*x^14 - 13*x^13 - x^12 + 6*x^11 + 10*x^10 - 26*x^9 + 20*x^8 - 6*x^7 + x^6 + 5*x^5 - 10*x^4 + 6*x^3 - 2*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 3 x^{19} \) \(\mathstrut +\mathstrut 2 x^{18} \) \(\mathstrut +\mathstrut 3 x^{17} \) \(\mathstrut -\mathstrut 2 x^{16} \) \(\mathstrut -\mathstrut 9 x^{15} \) \(\mathstrut +\mathstrut 18 x^{14} \) \(\mathstrut -\mathstrut 13 x^{13} \) \(\mathstrut -\mathstrut x^{12} \) \(\mathstrut +\mathstrut 6 x^{11} \) \(\mathstrut +\mathstrut 10 x^{10} \) \(\mathstrut -\mathstrut 26 x^{9} \) \(\mathstrut +\mathstrut 20 x^{8} \) \(\mathstrut -\mathstrut 6 x^{7} \) \(\mathstrut +\mathstrut x^{6} \) \(\mathstrut +\mathstrut 5 x^{5} \) \(\mathstrut -\mathstrut 10 x^{4} \) \(\mathstrut +\mathstrut 6 x^{3} \) \(\mathstrut -\mathstrut 2 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(3575480621700351753081=3^{10}\cdot 7^{10}\cdot 11^{8}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.96$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2663593} a^{19} - \frac{605577}{2663593} a^{18} + \frac{865553}{2663593} a^{17} - \frac{1243914}{2663593} a^{16} - \frac{768917}{2663593} a^{15} + \frac{133054}{2663593} a^{14} - \frac{354728}{2663593} a^{13} + \frac{605595}{2663593} a^{12} - \frac{1111512}{2663593} a^{11} - \frac{501171}{2663593} a^{10} + \frac{1013558}{2663593} a^{9} + \frac{680637}{2663593} a^{8} - \frac{1035426}{2663593} a^{7} + \frac{1290760}{2663593} a^{6} - \frac{685238}{2663593} a^{5} + \frac{1163147}{2663593} a^{4} - \frac{394096}{2663593} a^{3} - \frac{978097}{2663593} a^{2} - \frac{1053511}{2663593} a + \frac{402138}{2663593}$

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:  \( -\frac{5069577}{2663593} a^{19} + \frac{15210989}{2663593} a^{18} - \frac{13772404}{2663593} a^{17} - \frac{5212947}{2663593} a^{16} + \frac{3481178}{2663593} a^{15} + \frac{38083464}{2663593} a^{14} - \frac{89138870}{2663593} a^{13} + \frac{95969693}{2663593} a^{12} - \frac{52519168}{2663593} a^{11} + \frac{16837722}{2663593} a^{10} - \frac{63630828}{2663593} a^{9} + \frac{132396579}{2663593} a^{8} - \frac{146687436}{2663593} a^{7} + \frac{116295184}{2663593} a^{6} - \frac{79084843}{2663593} a^{5} + \frac{20213325}{2663593} a^{4} + \frac{18815882}{2663593} a^{3} - \frac{27982575}{2663593} a^{2} + \frac{18220722}{2663593} a - \frac{3418100}{2663593} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{2748078}{2663593} a^{19} - \frac{10532873}{2663593} a^{18} + \frac{13280948}{2663593} a^{17} - \frac{12475}{2663593} a^{16} - \frac{7573847}{2663593} a^{15} - \frac{20604014}{2663593} a^{14} + \frac{68143181}{2663593} a^{13} - \frac{83835745}{2663593} a^{12} + \frac{49488162}{2663593} a^{11} - \frac{11611979}{2663593} a^{10} + \frac{35886575}{2663593} a^{9} - \frac{101908866}{2663593} a^{8} + \frac{127608160}{2663593} a^{7} - \frac{96191761}{2663593} a^{6} + \frac{62124064}{2663593} a^{5} - \frac{28761777}{2663593} a^{4} - \frac{8278839}{2663593} a^{3} + \frac{27420117}{2663593} a^{2} - \frac{15234705}{2663593} a + \frac{3163808}{2663593} \),  \( \frac{3844207}{2663593} a^{19} - \frac{12359962}{2663593} a^{18} + \frac{12516650}{2663593} a^{17} + \frac{3035319}{2663593} a^{16} - \frac{4390336}{2663593} a^{15} - \frac{29281542}{2663593} a^{14} + \frac{74465002}{2663593} a^{13} - \frac{83587078}{2663593} a^{12} + \frac{48352430}{2663593} a^{11} - \frac{14931532}{2663593} a^{10} + \frac{49554036}{2663593} a^{9} - \frac{111017586}{2663593} a^{8} + \frac{129446692}{2663593} a^{7} - \frac{101389868}{2663593} a^{6} + \frac{70124804}{2663593} a^{5} - \frac{22365601}{2663593} a^{4} - \frac{14811262}{2663593} a^{3} + \frac{24424848}{2663593} a^{2} - \frac{14430032}{2663593} a + \frac{5609226}{2663593} \),  \( \frac{3033068}{2663593} a^{19} - \frac{11740854}{2663593} a^{18} + \frac{14546281}{2663593} a^{17} - \frac{143779}{2663593} a^{16} - \frac{7433567}{2663593} a^{15} - \frac{23118095}{2663593} a^{14} + \frac{73546369}{2663593} a^{13} - \frac{93479502}{2663593} a^{12} + \frac{58135179}{2663593} a^{11} - \frac{15815016}{2663593} a^{10} + \frac{39101703}{2663593} a^{9} - \frac{111321240}{2663593} a^{8} + \frac{141020890}{2663593} a^{7} - \frac{113992184}{2663593} a^{6} + \frac{73448797}{2663593} a^{5} - \frac{30310690}{2663593} a^{4} - \frac{11299034}{2663593} a^{3} + \frac{27227130}{2663593} a^{2} - \frac{20458821}{2663593} a + \frac{4720010}{2663593} \),  \( \frac{5409575}{2663593} a^{19} - \frac{17104528}{2663593} a^{18} + \frac{15652286}{2663593} a^{17} + \frac{7564501}{2663593} a^{16} - \frac{7397580}{2663593} a^{15} - \frac{43780270}{2663593} a^{14} + \frac{103137331}{2663593} a^{13} - \frac{103937748}{2663593} a^{12} + \frac{45909846}{2663593} a^{11} - \frac{7949612}{2663593} a^{10} + \frac{68979337}{2663593} a^{9} - \frac{154110030}{2663593} a^{8} + \frac{158995877}{2663593} a^{7} - \frac{108732198}{2663593} a^{6} + \frac{73360657}{2663593} a^{5} - \frac{21539515}{2663593} a^{4} - \frac{26612604}{2663593} a^{3} + \frac{34535598}{2663593} a^{2} - \frac{15194246}{2663593} a + \frac{1977948}{2663593} \),  \( \frac{8939390}{2663593} a^{19} - \frac{23873574}{2663593} a^{18} + \frac{12563854}{2663593} a^{17} + \frac{23813034}{2663593} a^{16} - \frac{4817167}{2663593} a^{15} - \frac{77774101}{2663593} a^{14} + \frac{133991311}{2663593} a^{13} - \frac{92231078}{2663593} a^{12} + \frac{1433590}{2663593} a^{11} + \frac{16381868}{2663593} a^{10} + \frac{109637227}{2663593} a^{9} - \frac{200294654}{2663593} a^{8} + \frac{143241044}{2663593} a^{7} - \frac{67185228}{2663593} a^{6} + \frac{44212383}{2663593} a^{5} + \frac{17928869}{2663593} a^{4} - \frac{56467780}{2663593} a^{3} + \frac{31711876}{2663593} a^{2} - \frac{2683993}{2663593} a - \frac{7259142}{2663593} \),  \( \frac{1535220}{2663593} a^{19} - \frac{6739185}{2663593} a^{18} + \frac{11655192}{2663593} a^{17} - \frac{5331765}{2663593} a^{16} - \frac{5611000}{2663593} a^{15} - \frac{6448883}{2663593} a^{14} + \frac{47331329}{2663593} a^{13} - \frac{74834168}{2663593} a^{12} + \frac{58020298}{2663593} a^{11} - \frac{18250198}{2663593} a^{10} + \frac{16754020}{2663593} a^{9} - \frac{69252178}{2663593} a^{8} + \frac{116160842}{2663593} a^{7} - \frac{99306835}{2663593} a^{6} + \frac{58899222}{2663593} a^{5} - \frac{31490941}{2663593} a^{4} + \frac{3098051}{2663593} a^{3} + \frac{22458968}{2663593} a^{2} - \frac{18842669}{2663593} a + \frac{5378413}{2663593} \),  \( \frac{3632131}{2663593} a^{19} - \frac{9148605}{2663593} a^{18} + \frac{5683031}{2663593} a^{17} + \frac{5090063}{2663593} a^{16} + \frac{1951489}{2663593} a^{15} - \frac{26073811}{2663593} a^{14} + \frac{49267301}{2663593} a^{13} - \frac{47998315}{2663593} a^{12} + \frac{24082098}{2663593} a^{11} - \frac{11278422}{2663593} a^{10} + \frac{44855542}{2663593} a^{9} - \frac{72404247}{2663593} a^{8} + \frac{73245102}{2663593} a^{7} - \frac{63201123}{2663593} a^{6} + \frac{43052475}{2663593} a^{5} - \frac{3935892}{2663593} a^{4} - \frac{9401934}{2663593} a^{3} + \frac{14701380}{2663593} a^{2} - \frac{12212036}{2663593} a + \frac{2047819}{2663593} \),  \( \frac{74723}{2663593} a^{19} - \frac{4075880}{2663593} a^{18} + \frac{10005965}{2663593} a^{17} - \frac{5571680}{2663593} a^{16} - \frac{7411167}{2663593} a^{15} - \frac{998627}{2663593} a^{14} + \frac{31036715}{2663593} a^{13} - \frac{55841745}{2663593} a^{12} + \frac{45926831}{2663593} a^{11} - \frac{14864611}{2663593} a^{10} + \frac{4818258}{2663593} a^{9} - \frac{50014458}{2663593} a^{8} + \frac{81820256}{2663593} a^{7} - \frac{73823654}{2663593} a^{6} + \frac{49817432}{2663593} a^{5} - \frac{33833018}{2663593} a^{4} - \frac{2014793}{2663593} a^{3} + \frac{13304161}{2663593} a^{2} - \frac{14992896}{2663593} a + \frac{6292327}{2663593} \),  \( \frac{461180}{2663593} a^{19} - \frac{4938403}{2663593} a^{18} + \frac{9685560}{2663593} a^{17} - \frac{2243331}{2663593} a^{16} - \frac{10333156}{2663593} a^{15} - \frac{4675407}{2663593} a^{14} + \frac{39049729}{2663593} a^{13} - \frac{53350182}{2663593} a^{12} + \frac{28004620}{2663593} a^{11} + \frac{3240795}{2663593} a^{10} + \frac{4070056}{2663593} a^{9} - \frac{58871657}{2663593} a^{8} + \frac{80443778}{2663593} a^{7} - \frac{48329479}{2663593} a^{6} + \frac{22575796}{2663593} a^{5} - \frac{19505961}{2663593} a^{4} - \frac{6915704}{2663593} a^{3} + \frac{22008834}{2663593} a^{2} - \frac{8185408}{2663593} a + \frac{13029}{2663593} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 635.552521857 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_{10}\times D_5$ (as 20T24):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 100
The 40 conjugacy class representatives for $C_{10}\times D_5$
Character table for $C_{10}\times D_5$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}, \sqrt{-7})\), 10.0.246071287.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
$7$7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$