Properties

Label 20.0.20865144565...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{8}\cdot 67^{6}$
Root discriminant $11.64$
Ramified primes $3, 5, 67$
Class number $1$
Class group Trivial
Galois Group 20T288

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 4, -11, 22, -24, 35, -47, 42, -52, 65, -62, 64, -60, 45, -29, 19, -14, 9, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 9*x^18 - 14*x^17 + 19*x^16 - 29*x^15 + 45*x^14 - 60*x^13 + 64*x^12 - 62*x^11 + 65*x^10 - 52*x^9 + 42*x^8 - 47*x^7 + 35*x^6 - 24*x^5 + 22*x^4 - 11*x^3 + 4*x^2 - 3*x + 1)
gp: K = bnfinit(x^20 - 4*x^19 + 9*x^18 - 14*x^17 + 19*x^16 - 29*x^15 + 45*x^14 - 60*x^13 + 64*x^12 - 62*x^11 + 65*x^10 - 52*x^9 + 42*x^8 - 47*x^7 + 35*x^6 - 24*x^5 + 22*x^4 - 11*x^3 + 4*x^2 - 3*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 4 x^{19} \) \(\mathstrut +\mathstrut 9 x^{18} \) \(\mathstrut -\mathstrut 14 x^{17} \) \(\mathstrut +\mathstrut 19 x^{16} \) \(\mathstrut -\mathstrut 29 x^{15} \) \(\mathstrut +\mathstrut 45 x^{14} \) \(\mathstrut -\mathstrut 60 x^{13} \) \(\mathstrut +\mathstrut 64 x^{12} \) \(\mathstrut -\mathstrut 62 x^{11} \) \(\mathstrut +\mathstrut 65 x^{10} \) \(\mathstrut -\mathstrut 52 x^{9} \) \(\mathstrut +\mathstrut 42 x^{8} \) \(\mathstrut -\mathstrut 47 x^{7} \) \(\mathstrut +\mathstrut 35 x^{6} \) \(\mathstrut -\mathstrut 24 x^{5} \) \(\mathstrut +\mathstrut 22 x^{4} \) \(\mathstrut -\mathstrut 11 x^{3} \) \(\mathstrut +\mathstrut 4 x^{2} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(2086514456522375390625=3^{10}\cdot 5^{8}\cdot 67^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.64$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 5, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{21} a^{17} + \frac{1}{7} a^{16} + \frac{3}{7} a^{15} + \frac{10}{21} a^{14} + \frac{1}{3} a^{13} - \frac{3}{7} a^{12} + \frac{5}{21} a^{11} - \frac{4}{21} a^{10} + \frac{2}{21} a^{9} - \frac{5}{21} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{3}{7} a^{3} + \frac{8}{21} a^{2} + \frac{1}{21} a - \frac{5}{21}$, $\frac{1}{105} a^{18} - \frac{1}{15} a^{16} - \frac{52}{105} a^{15} + \frac{11}{35} a^{14} - \frac{2}{7} a^{13} - \frac{38}{105} a^{12} - \frac{4}{35} a^{11} + \frac{2}{15} a^{10} - \frac{4}{105} a^{9} - \frac{2}{7} a^{8} + \frac{47}{105} a^{7} + \frac{5}{21} a^{6} - \frac{4}{105} a^{5} - \frac{26}{105} a^{4} + \frac{16}{105} a^{3} + \frac{47}{105} a^{2} - \frac{29}{105} a - \frac{34}{105}$, $\frac{1}{207795} a^{19} - \frac{338}{207795} a^{18} + \frac{14}{29685} a^{17} - \frac{7617}{69265} a^{16} + \frac{23498}{69265} a^{15} - \frac{66169}{207795} a^{14} - \frac{86078}{207795} a^{13} - \frac{1626}{69265} a^{12} - \frac{4050}{13853} a^{11} - \frac{55661}{207795} a^{10} - \frac{22391}{69265} a^{9} - \frac{89248}{207795} a^{8} - \frac{103711}{207795} a^{7} - \frac{60359}{207795} a^{6} - \frac{15964}{207795} a^{5} - \frac{14997}{69265} a^{4} - \frac{49006}{207795} a^{3} + \frac{7457}{41559} a^{2} + \frac{18691}{69265} a - \frac{70373}{207795}$

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:  \( \frac{446380}{41559} a^{19} - \frac{507763}{13853} a^{18} + \frac{3138139}{41559} a^{17} - \frac{637280}{5937} a^{16} + \frac{5973073}{41559} a^{15} - \frac{1369838}{5937} a^{14} + \frac{4883015}{13853} a^{13} - \frac{18497107}{41559} a^{12} + \frac{18213646}{41559} a^{11} - \frac{5870395}{13853} a^{10} + \frac{19242005}{41559} a^{9} - \frac{12423695}{41559} a^{8} + \frac{574441}{1979} a^{7} - \frac{4774813}{13853} a^{6} + \frac{2508248}{13853} a^{5} - \frac{6669826}{41559} a^{4} + \frac{6152458}{41559} a^{3} - \frac{477231}{13853} a^{2} + \frac{1074751}{41559} a - \frac{747401}{41559} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{2899322}{207795} a^{19} - \frac{9845246}{207795} a^{18} + \frac{20174536}{207795} a^{17} - \frac{28457162}{207795} a^{16} + \frac{37905823}{207795} a^{15} - \frac{61020278}{207795} a^{14} + \frac{93281894}{207795} a^{13} - \frac{117304126}{207795} a^{12} + \frac{7619062}{13853} a^{11} - \frac{36592994}{69265} a^{10} + \frac{120386464}{207795} a^{9} - \frac{76059866}{207795} a^{8} + \frac{74263358}{207795} a^{7} - \frac{89398313}{207795} a^{6} + \frac{6564116}{29685} a^{5} - \frac{41089732}{207795} a^{4} + \frac{1783043}{9895} a^{3} - \frac{537746}{13853} a^{2} + \frac{6174596}{207795} a - \frac{1496592}{69265} \),  \( \frac{1317224}{207795} a^{19} - \frac{4516223}{207795} a^{18} + \frac{9249607}{207795} a^{17} - \frac{13083317}{207795} a^{16} + \frac{17427853}{207795} a^{15} - \frac{4022147}{29685} a^{14} + \frac{43179023}{207795} a^{13} - \frac{7760152}{29685} a^{12} + \frac{17691524}{69265} a^{11} - \frac{17050801}{69265} a^{10} + \frac{8096341}{29685} a^{9} - \frac{36870662}{207795} a^{8} + \frac{34932419}{207795} a^{7} - \frac{6071063}{29685} a^{6} + \frac{21936608}{207795} a^{5} - \frac{19318948}{207795} a^{4} + \frac{1214813}{13853} a^{3} - \frac{200902}{9895} a^{2} + \frac{445313}{29685} a - \frac{778206}{69265} \),  \( \frac{8769}{1979} a^{19} - \frac{644008}{41559} a^{18} + \frac{1349779}{41559} a^{17} - \frac{1952813}{41559} a^{16} + \frac{2616919}{41559} a^{15} - \frac{592973}{5937} a^{14} + \frac{6363352}{41559} a^{13} - \frac{8142433}{41559} a^{12} + \frac{8149271}{41559} a^{11} - \frac{2612830}{13853} a^{10} + \frac{2817980}{13853} a^{9} - \frac{5738996}{41559} a^{8} + \frac{5407840}{41559} a^{7} - \frac{6225883}{41559} a^{6} + \frac{497350}{5937} a^{5} - \frac{3001924}{41559} a^{4} + \frac{2673971}{41559} a^{3} - \frac{240308}{13853} a^{2} + \frac{161004}{13853} a - \frac{48940}{5937} \),  \( \frac{1444658}{207795} a^{19} - \frac{4795019}{207795} a^{18} + \frac{9666239}{207795} a^{17} - \frac{13361098}{207795} a^{16} + \frac{5912294}{69265} a^{15} - \frac{9650114}{69265} a^{14} + \frac{14751432}{69265} a^{13} - \frac{2617974}{9895} a^{12} + \frac{1497728}{5937} a^{11} - \frac{7204264}{29685} a^{10} + \frac{8056598}{29685} a^{9} - \frac{34323439}{207795} a^{8} + \frac{11596194}{69265} a^{7} - \frac{14046944}{69265} a^{6} + \frac{973178}{9895} a^{5} - \frac{2741534}{29685} a^{4} + \frac{17548102}{207795} a^{3} - \frac{97315}{5937} a^{2} + \frac{992793}{69265} a - \frac{2282344}{207795} \),  \( \frac{1048673}{207795} a^{19} - \frac{520577}{29685} a^{18} + \frac{2529983}{69265} a^{17} - \frac{520693}{9895} a^{16} + \frac{2094251}{29685} a^{15} - \frac{23299147}{207795} a^{14} + \frac{35569861}{207795} a^{13} - \frac{6459857}{29685} a^{12} + \frac{9017263}{41559} a^{11} - \frac{43477073}{207795} a^{10} + \frac{2228896}{9895} a^{9} - \frac{1456524}{9895} a^{8} + \frac{28822027}{207795} a^{7} - \frac{34675147}{207795} a^{6} + \frac{18779008}{207795} a^{5} - \frac{5307916}{69265} a^{4} + \frac{2163701}{29685} a^{3} - \frac{99320}{5937} a^{2} + \frac{2292079}{207795} a - \frac{1988309}{207795} \),  \( \frac{267857}{69265} a^{19} - \frac{2633192}{207795} a^{18} + \frac{5378533}{207795} a^{17} - \frac{1081109}{29685} a^{16} + \frac{3421189}{69265} a^{15} - \frac{5528172}{69265} a^{14} + \frac{3581401}{29685} a^{13} - \frac{10478207}{69265} a^{12} + \frac{10283781}{69265} a^{11} - \frac{30393887}{207795} a^{10} + \frac{33145358}{207795} a^{9} - \frac{20368778}{207795} a^{8} + \frac{21687151}{207795} a^{7} - \frac{3466777}{29685} a^{6} + \frac{1736581}{29685} a^{5} - \frac{12210862}{207795} a^{4} + \frac{703335}{13853} a^{3} - \frac{2215468}{207795} a^{2} + \frac{312512}{29685} a - \frac{1223632}{207795} \),  \( \frac{365954}{29685} a^{19} - \frac{8684873}{207795} a^{18} + \frac{5913378}{69265} a^{17} - \frac{4998884}{41559} a^{16} + \frac{2223355}{13853} a^{15} - \frac{53920049}{207795} a^{14} + \frac{82434571}{207795} a^{13} - \frac{34480169}{69265} a^{12} + \frac{4797933}{9895} a^{11} - \frac{97290014}{207795} a^{10} + \frac{107494627}{207795} a^{9} - \frac{3251694}{9895} a^{8} + \frac{22134648}{69265} a^{7} - \frac{3819167}{9895} a^{6} + \frac{13566158}{69265} a^{5} - \frac{36844124}{207795} a^{4} + \frac{34191013}{207795} a^{3} - \frac{2529851}{69265} a^{2} + \frac{172262}{5937} a - \frac{1395206}{69265} \),  \( \frac{66007}{9895} a^{19} - \frac{4735078}{207795} a^{18} + \frac{3246842}{69265} a^{17} - \frac{4607171}{69265} a^{16} + \frac{18477932}{207795} a^{15} - \frac{29672989}{207795} a^{14} + \frac{15105798}{69265} a^{13} - \frac{11412400}{41559} a^{12} + \frac{55966849}{207795} a^{11} - \frac{10799233}{41559} a^{10} + \frac{59038247}{207795} a^{9} - \frac{12528787}{69265} a^{8} + \frac{12000208}{69265} a^{7} - \frac{14511536}{69265} a^{6} + \frac{1525084}{13853} a^{5} - \frac{1330927}{13853} a^{4} + \frac{2680903}{29685} a^{3} - \frac{4251379}{207795} a^{2} + \frac{3098924}{207795} a - \frac{736991}{69265} \),  \( \frac{2057777}{207795} a^{19} - \frac{65695}{1979} a^{18} + \frac{14112316}{207795} a^{17} - \frac{2837207}{29685} a^{16} + \frac{1266951}{9895} a^{15} - \frac{2861921}{13853} a^{14} + \frac{65423474}{207795} a^{13} - \frac{3911159}{9895} a^{12} + \frac{80058613}{207795} a^{11} - \frac{77550698}{207795} a^{10} + \frac{810337}{1979} a^{9} - \frac{7615658}{29685} a^{8} + \frac{10694281}{41559} a^{7} - \frac{62523728}{207795} a^{6} + \frac{32017343}{207795} a^{5} - \frac{29639083}{207795} a^{4} + \frac{3738397}{29685} a^{3} - \frac{841879}{29685} a^{2} + \frac{4793287}{207795} a - \frac{197447}{13853} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 527.926443922 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

20T288:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n288
Character table for t20n288 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.1.5025.1, 10.0.681766875.1, 10.0.1691791875.1, 10.2.45678380625.1

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 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
67Data not computed