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Properties

Label 20.4.87082070223...4081.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{4}\cdot 401^{10}$
Root discriminant $24.95$
Ramified primes $3, 401$
Class number $1$
Class group Trivial
Galois group $C_2^4:D_5$ (as 20T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-32, 400, -1728, 4008, -5594, 4777, -2385, 1233, -1379, -91, 2133, -1563, -141, 460, -54, -42, -8, 1, 11, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 11*x^18 + x^17 - 8*x^16 - 42*x^15 - 54*x^14 + 460*x^13 - 141*x^12 - 1563*x^11 + 2133*x^10 - 91*x^9 - 1379*x^8 + 1233*x^7 - 2385*x^6 + 4777*x^5 - 5594*x^4 + 4008*x^3 - 1728*x^2 + 400*x - 32)
 
gp: K = bnfinit(x^20 - 6*x^19 + 11*x^18 + x^17 - 8*x^16 - 42*x^15 - 54*x^14 + 460*x^13 - 141*x^12 - 1563*x^11 + 2133*x^10 - 91*x^9 - 1379*x^8 + 1233*x^7 - 2385*x^6 + 4777*x^5 - 5594*x^4 + 4008*x^3 - 1728*x^2 + 400*x - 32, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 11 x^{18} + x^{17} - 8 x^{16} - 42 x^{15} - 54 x^{14} + 460 x^{13} - 141 x^{12} - 1563 x^{11} + 2133 x^{10} - 91 x^{9} - 1379 x^{8} + 1233 x^{7} - 2385 x^{6} + 4777 x^{5} - 5594 x^{4} + 4008 x^{3} - 1728 x^{2} + 400 x - 32 \)

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:  \(8708207022330414958663524081=3^{4}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.95$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{12} a^{16} - \frac{1}{6} a^{15} - \frac{1}{12} a^{14} - \frac{1}{4} a^{13} - \frac{1}{3} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} + \frac{1}{3} a^{9} - \frac{1}{12} a^{8} + \frac{5}{12} a^{7} - \frac{1}{4} a^{6} + \frac{5}{12} a^{5} + \frac{5}{12} a^{4} - \frac{1}{4} a^{3} - \frac{5}{12} a^{2} - \frac{1}{4} a + \frac{1}{6}$, $\frac{1}{16176} a^{17} + \frac{23}{674} a^{16} + \frac{655}{16176} a^{15} + \frac{1483}{16176} a^{14} + \frac{3895}{8088} a^{13} + \frac{1775}{8088} a^{12} - \frac{731}{2696} a^{11} + \frac{137}{2022} a^{10} + \frac{2875}{16176} a^{9} - \frac{1339}{5392} a^{8} + \frac{3523}{16176} a^{7} + \frac{7715}{16176} a^{6} + \frac{905}{5392} a^{5} - \frac{6257}{16176} a^{4} - \frac{2315}{16176} a^{3} - \frac{7549}{16176} a^{2} + \frac{1979}{4044} a - \frac{839}{4044}$, $\frac{1}{194112} a^{18} + \frac{1}{97056} a^{17} + \frac{4399}{194112} a^{16} - \frac{8287}{194112} a^{15} - \frac{573}{5392} a^{14} - \frac{5243}{97056} a^{13} + \frac{29861}{97056} a^{12} + \frac{22361}{48528} a^{11} + \frac{9371}{194112} a^{10} - \frac{3601}{64704} a^{9} - \frac{46375}{194112} a^{8} - \frac{3461}{64704} a^{7} + \frac{8179}{64704} a^{6} - \frac{21745}{64704} a^{5} + \frac{19417}{64704} a^{4} + \frac{3973}{194112} a^{3} - \frac{1651}{10784} a^{2} + \frac{5287}{48528} a + \frac{4261}{24264}$, $\frac{1}{441047127087283968} a^{19} + \frac{60506188511}{110261781771820992} a^{18} + \frac{13007817162019}{441047127087283968} a^{17} - \frac{18265952741528185}{441047127087283968} a^{16} + \frac{9446662762226065}{73507854514547328} a^{15} + \frac{21423324662675}{157180016780928} a^{14} + \frac{90260905880425127}{220523563543641984} a^{13} + \frac{19023278669231797}{55130890885910496} a^{12} + \frac{157598839609794563}{441047127087283968} a^{11} - \frac{11414724119079943}{147015709029094656} a^{10} + \frac{133880204856438539}{441047127087283968} a^{9} - \frac{52526368809902903}{147015709029094656} a^{8} + \frac{2295357993039191}{6391987349091072} a^{7} - \frac{52708661915952227}{147015709029094656} a^{6} + \frac{2484558710441177}{6391987349091072} a^{5} - \frac{108424263632124605}{441047127087283968} a^{4} + \frac{14738183493528769}{36753927257273664} a^{3} + \frac{312367923861983}{13782722721477624} a^{2} - \frac{1524842891467357}{6891361360738812} a - \frac{1347814544555727}{3062827271439472}$

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

Class group and class number

Trivial group, which has 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:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1169621.82282 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:D_5$ (as 20T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 10 conjugacy class representatives for $C_2^4:D_5$
Character table for $C_2^4:D_5$

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.2.232712654409.1, 10.2.93317774418009.3, 10.10.10368641602001.1

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 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
401Data not computed