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Properties

Label 20.4.13289445530...5625.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{12}\cdot 5^{12}\cdot 97^{2}\cdot 691^{6}$
Root discriminant $57.04$
Ramified primes $3, 5, 97, 691$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![53181, -42039, -180810, 586329, 236898, -41223, -182609, 207092, 27394, 2665, -47926, 25126, -1148, 2783, 1640, -1551, 369, -118, 20, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 20*x^18 - 118*x^17 + 369*x^16 - 1551*x^15 + 1640*x^14 + 2783*x^13 - 1148*x^12 + 25126*x^11 - 47926*x^10 + 2665*x^9 + 27394*x^8 + 207092*x^7 - 182609*x^6 - 41223*x^5 + 236898*x^4 + 586329*x^3 - 180810*x^2 - 42039*x + 53181)
 
gp: K = bnfinit(x^20 - 5*x^19 + 20*x^18 - 118*x^17 + 369*x^16 - 1551*x^15 + 1640*x^14 + 2783*x^13 - 1148*x^12 + 25126*x^11 - 47926*x^10 + 2665*x^9 + 27394*x^8 + 207092*x^7 - 182609*x^6 - 41223*x^5 + 236898*x^4 + 586329*x^3 - 180810*x^2 - 42039*x + 53181, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 20 x^{18} - 118 x^{17} + 369 x^{16} - 1551 x^{15} + 1640 x^{14} + 2783 x^{13} - 1148 x^{12} + 25126 x^{11} - 47926 x^{10} + 2665 x^{9} + 27394 x^{8} + 207092 x^{7} - 182609 x^{6} - 41223 x^{5} + 236898 x^{4} + 586329 x^{3} - 180810 x^{2} - 42039 x + 53181 \)

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:  \(132894455303943784655361213134765625=3^{12}\cdot 5^{12}\cdot 97^{2}\cdot 691^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.04$
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:  $3, 5, 97, 691$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{405471390726179944101644357987593789956236341958908982216006237829} a^{19} - \frac{58410232379520043079706829964197525367616183188334338516857968289}{405471390726179944101644357987593789956236341958908982216006237829} a^{18} - \frac{47202128577815253092621052606359334330745782832742930496097592758}{405471390726179944101644357987593789956236341958908982216006237829} a^{17} - \frac{28231349879748000469801170274634745420162970580902549615133568349}{405471390726179944101644357987593789956236341958908982216006237829} a^{16} - \frac{176025520445865876560968798133773884002847742455731326097004641}{7113533170634735861432357157677084034319935823840508459929933997} a^{15} + \frac{124935365300524654117071567866585645188356309678443758272351869485}{405471390726179944101644357987593789956236341958908982216006237829} a^{14} + \frac{523267853292161059376065516584580541816558694383514281976403214}{135157130242059981367214785995864596652078780652969660738668745943} a^{13} - \frac{4294790571753066853968928173759630891145120683358907264518933345}{135157130242059981367214785995864596652078780652969660738668745943} a^{12} + \frac{44559243909050100805269173858517910487029927960359638940767662703}{135157130242059981367214785995864596652078780652969660738668745943} a^{11} + \frac{143133783629090396313411329524261964140665288050558340304110491410}{405471390726179944101644357987593789956236341958908982216006237829} a^{10} - \frac{146182521450606068194436150329515724255123630148534148162442146866}{405471390726179944101644357987593789956236341958908982216006237829} a^{9} - \frac{63547738804306385780130843138463874185199646859453526499540350158}{135157130242059981367214785995864596652078780652969660738668745943} a^{8} - \frac{41628853517856166862542110644268316558040259980433799804116948133}{135157130242059981367214785995864596652078780652969660738668745943} a^{7} + \frac{90643363372053763559690961605477034545394243016044220362571663537}{405471390726179944101644357987593789956236341958908982216006237829} a^{6} - \frac{147137330966154863012762160427509304339852445825089524396783875953}{405471390726179944101644357987593789956236341958908982216006237829} a^{5} - \frac{17185927894417532358918689357085873692252174790201419619600553040}{135157130242059981367214785995864596652078780652969660738668745943} a^{4} - \frac{14408615745047762676452550548671646780691964369256332449624233998}{405471390726179944101644357987593789956236341958908982216006237829} a^{3} + \frac{62322039038922787892817058149144183371767855385350012802921239238}{135157130242059981367214785995864596652078780652969660738668745943} a^{2} - \frac{28520988947274208915055138947920547934632277851059875449286347978}{135157130242059981367214785995864596652078780652969660738668745943} a - \frac{2434157271113345571688846958057657350520015023145321724031673479}{7113533170634735861432357157677084034319935823840508459929933997}$

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:  \( 4962031716.78 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.5438807015625.1

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
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 $16{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ R R $16{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ $16{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.6.6.5$x^{6} + 6 x^{3} + 9 x^{2} + 9$$3$$2$$6$$S_3^2$$[3/2, 3/2]_{2}^{2}$
3.6.6.5$x^{6} + 6 x^{3} + 9 x^{2} + 9$$3$$2$$6$$S_3^2$$[3/2, 3/2]_{2}^{2}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$97$97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
691Data not computed