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Properties

Label 20.4.35876495739...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 5^{10}\cdot 89^{9}$
Root discriminant $33.71$
Ramified primes $2, 5, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T658

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, 512, -2048, -4480, -4544, 640, 10096, 8504, -920, 218, 2841, 1905, -1273, 1172, -157, -35, 83, -22, 9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 9*x^18 - 22*x^17 + 83*x^16 - 35*x^15 - 157*x^14 + 1172*x^13 - 1273*x^12 + 1905*x^11 + 2841*x^10 + 218*x^9 - 920*x^8 + 8504*x^7 + 10096*x^6 + 640*x^5 - 4544*x^4 - 4480*x^3 - 2048*x^2 + 512*x + 1024)
 
gp: K = bnfinit(x^20 - x^19 + 9*x^18 - 22*x^17 + 83*x^16 - 35*x^15 - 157*x^14 + 1172*x^13 - 1273*x^12 + 1905*x^11 + 2841*x^10 + 218*x^9 - 920*x^8 + 8504*x^7 + 10096*x^6 + 640*x^5 - 4544*x^4 - 4480*x^3 - 2048*x^2 + 512*x + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 9 x^{18} - 22 x^{17} + 83 x^{16} - 35 x^{15} - 157 x^{14} + 1172 x^{13} - 1273 x^{12} + 1905 x^{11} + 2841 x^{10} + 218 x^{9} - 920 x^{8} + 8504 x^{7} + 10096 x^{6} + 640 x^{5} - 4544 x^{4} - 4480 x^{3} - 2048 x^{2} + 512 x + 1024 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3587649573964648540160000000000=2^{20}\cdot 5^{10}\cdot 89^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.71$
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:  $2, 5, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{3}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} + \frac{1}{16} a^{12} - \frac{1}{8} a^{11} - \frac{1}{16} a^{10} - \frac{3}{16} a^{9} + \frac{3}{16} a^{8} + \frac{3}{16} a^{6} - \frac{7}{16} a^{5} + \frac{1}{16} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{14} + \frac{1}{32} a^{13} + \frac{1}{16} a^{12} + \frac{3}{32} a^{11} - \frac{3}{32} a^{10} - \frac{5}{32} a^{9} - \frac{1}{8} a^{8} + \frac{7}{32} a^{7} - \frac{15}{32} a^{6} - \frac{15}{32} a^{5} + \frac{1}{16} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{16} - \frac{1}{64} a^{15} + \frac{1}{64} a^{14} + \frac{1}{32} a^{13} - \frac{5}{64} a^{12} - \frac{3}{64} a^{11} - \frac{5}{64} a^{10} - \frac{1}{16} a^{9} - \frac{1}{64} a^{8} + \frac{1}{64} a^{7} - \frac{15}{64} a^{6} - \frac{15}{32} a^{5} + \frac{1}{4} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{256} a^{17} + \frac{1}{256} a^{16} - \frac{1}{256} a^{15} + \frac{1}{64} a^{14} + \frac{15}{256} a^{13} - \frac{29}{256} a^{12} + \frac{21}{256} a^{11} - \frac{7}{128} a^{10} - \frac{25}{256} a^{9} - \frac{17}{256} a^{8} - \frac{13}{256} a^{7} + \frac{9}{64} a^{6} - \frac{23}{64} a^{5} + \frac{1}{32} a^{4} + \frac{7}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10240} a^{18} + \frac{11}{10240} a^{17} - \frac{11}{2048} a^{16} + \frac{21}{5120} a^{15} + \frac{167}{10240} a^{14} - \frac{183}{10240} a^{13} + \frac{151}{2048} a^{12} - \frac{15}{512} a^{11} + \frac{427}{10240} a^{10} - \frac{411}{10240} a^{9} - \frac{1943}{10240} a^{8} + \frac{409}{5120} a^{7} + \frac{1031}{2560} a^{6} - \frac{1}{40} a^{5} + \frac{1}{4} a^{4} - \frac{13}{80} a^{3} + \frac{17}{160} a^{2} + \frac{3}{16} a + \frac{9}{40}$, $\frac{1}{58660372987559910181049647554560} a^{19} - \frac{141872284474285415430083351}{11732074597511982036209929510912} a^{18} - \frac{97735356274082627623810209121}{58660372987559910181049647554560} a^{17} - \frac{95723622586117504031106865457}{14665093246889977545262411888640} a^{16} - \frac{109749927416250094382617061633}{11732074597511982036209929510912} a^{15} - \frac{198259494685643986170109409237}{11732074597511982036209929510912} a^{14} - \frac{3115389943851824251985155204027}{58660372987559910181049647554560} a^{13} + \frac{8270326635524398464463355065}{5866037298755991018104964755456} a^{12} + \frac{847987287037070296765694828627}{58660372987559910181049647554560} a^{11} - \frac{309093174965134252499107454213}{58660372987559910181049647554560} a^{10} + \frac{14490634481202020733918302237043}{58660372987559910181049647554560} a^{9} + \frac{2806615944102855407694953941729}{14665093246889977545262411888640} a^{8} - \frac{35951514698773874132997085629}{229142081982655899144725185760} a^{7} - \frac{14506647646514302181164563341}{1466509324688997754526241188864} a^{6} - \frac{119164220899050774950974653823}{458284163965311798289450371520} a^{5} + \frac{86728464294324459291246598547}{458284163965311798289450371520} a^{4} - \frac{1904930339636651826482928753}{48240438312138084030468460160} a^{3} - \frac{24781424244349178392754017407}{57285520495663974786181296440} a^{2} + \frac{7399974449755385716554072393}{28642760247831987393090648220} a - \frac{1747964766216933321059071007}{114571040991327949572362592880}$

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:  $11$
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:  \( 49324768.242 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T658:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 76 conjugacy class representatives for t20n658 are not computed
Character table for t20n658 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, 10.2.25347200000.1

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 24 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 R $20$ R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$89$89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.6.4.1$x^{6} + 1513 x^{3} + 1710936$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
89.6.3.2$x^{6} - 7921 x^{2} + 4934783$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$