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Properties

Label 20.0.85672046165...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 41^{5}\cdot 27517559^{2}$
Root discriminant $31.38$
Ramified primes $5, 41, 27517559$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 20T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, 512, 5888, 256, 12672, -1344, 17136, -2752, 14636, -2850, 7499, -816, 1705, 302, 209, 67, 72, -8, 13, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 13*x^18 - 8*x^17 + 72*x^16 + 67*x^15 + 209*x^14 + 302*x^13 + 1705*x^12 - 816*x^11 + 7499*x^10 - 2850*x^9 + 14636*x^8 - 2752*x^7 + 17136*x^6 - 1344*x^5 + 12672*x^4 + 256*x^3 + 5888*x^2 + 512*x + 1024)
 
gp: K = bnfinit(x^20 - x^19 + 13*x^18 - 8*x^17 + 72*x^16 + 67*x^15 + 209*x^14 + 302*x^13 + 1705*x^12 - 816*x^11 + 7499*x^10 - 2850*x^9 + 14636*x^8 - 2752*x^7 + 17136*x^6 - 1344*x^5 + 12672*x^4 + 256*x^3 + 5888*x^2 + 512*x + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 13 x^{18} - 8 x^{17} + 72 x^{16} + 67 x^{15} + 209 x^{14} + 302 x^{13} + 1705 x^{12} - 816 x^{11} + 7499 x^{10} - 2850 x^{9} + 14636 x^{8} - 2752 x^{7} + 17136 x^{6} - 1344 x^{5} + 12672 x^{4} + 256 x^{3} + 5888 x^{2} + 512 x + 1024 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(856720461657154802252744140625=5^{10}\cdot 41^{5}\cdot 27517559^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.38$
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:  $5, 41, 27517559$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} + \frac{3}{8} a^{8} + \frac{1}{8} a^{7} + \frac{1}{4} a^{6} - \frac{3}{8} a^{5} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} + \frac{1}{16} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{5}{16} a^{9} + \frac{1}{16} a^{8} - \frac{3}{8} a^{7} + \frac{5}{16} a^{6} - \frac{1}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{14} + \frac{1}{32} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{11} + \frac{11}{32} a^{10} + \frac{1}{32} a^{9} + \frac{5}{16} a^{8} + \frac{5}{32} a^{7} - \frac{1}{2} a^{6} - \frac{1}{32} a^{5} - \frac{5}{16} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{64} a^{16} - \frac{1}{64} a^{15} + \frac{1}{64} a^{14} - \frac{1}{16} a^{13} + \frac{1}{16} a^{12} + \frac{11}{64} a^{11} - \frac{31}{64} a^{10} - \frac{11}{32} a^{9} + \frac{5}{64} a^{8} + \frac{1}{4} a^{7} + \frac{31}{64} a^{6} + \frac{11}{32} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{17} - \frac{1}{128} a^{16} + \frac{1}{128} a^{15} - \frac{1}{32} a^{14} + \frac{1}{32} a^{13} + \frac{11}{128} a^{12} - \frac{31}{128} a^{11} + \frac{21}{64} a^{10} + \frac{5}{128} a^{9} + \frac{1}{8} a^{8} + \frac{31}{128} a^{7} + \frac{11}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{256} a^{18} - \frac{1}{256} a^{17} + \frac{1}{256} a^{16} - \frac{1}{64} a^{15} + \frac{1}{64} a^{14} + \frac{11}{256} a^{13} - \frac{31}{256} a^{12} + \frac{21}{128} a^{11} + \frac{5}{256} a^{10} - \frac{7}{16} a^{9} - \frac{97}{256} a^{8} + \frac{11}{128} a^{7} + \frac{15}{32} a^{6} - \frac{3}{8} a^{5} - \frac{3}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{317803852015279311876690785792} a^{19} + \frac{511852731478205625355610803}{317803852015279311876690785792} a^{18} + \frac{688025712823531016601724561}{317803852015279311876690785792} a^{17} - \frac{464355052261517366502781}{235759534135963881214162304} a^{16} + \frac{231929596802026679999080453}{39725481501909913984586348224} a^{15} + \frac{3407443470525443335812155867}{317803852015279311876690785792} a^{14} + \frac{17892352469552439559494642045}{317803852015279311876690785792} a^{13} + \frac{7955706085711543743754045669}{158901926007639655938345392896} a^{12} - \frac{70705399223799452382388352463}{317803852015279311876690785792} a^{11} + \frac{18887256972560796561930687347}{79450963003819827969172696448} a^{10} + \frac{116214000736313433981147181891}{317803852015279311876690785792} a^{9} - \frac{66477439135038034926119422031}{158901926007639655938345392896} a^{8} + \frac{87320633829524944882860487}{220085770093683733986627968} a^{7} - \frac{6424646433308040957762045865}{39725481501909913984586348224} a^{6} + \frac{9399095942128803464814198099}{19862740750954956992293174112} a^{5} - \frac{1744990615203787382555270869}{4965685187738739248073293528} a^{4} - \frac{386168857294313063354449563}{2482842593869369624036646764} a^{3} - \frac{303443947067678550714083662}{620710648467342406009161691} a^{2} - \frac{263848411324005437893657173}{620710648467342406009161691} a + \frac{144006177692592892462734773}{620710648467342406009161691}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $9$
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:  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:  \( 838972.996958 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 115200
The 119 conjugacy class representatives for t20n781 are not computed
Character table for t20n781 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.1025.1, 10.8.85992371875.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ R $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
27517559Data not computed