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Properties

Label 20.20.7303816416...9296.3
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 11^{16}\cdot 23^{6}$
Root discriminant $49.34$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:C_5$ (as 20T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13729, 89954, 104664, -304834, -527870, 443584, 846276, -459512, -662559, 353360, 263516, -166594, -46263, 41306, 1570, -4992, 379, 282, -38, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 38*x^18 + 282*x^17 + 379*x^16 - 4992*x^15 + 1570*x^14 + 41306*x^13 - 46263*x^12 - 166594*x^11 + 263516*x^10 + 353360*x^9 - 662559*x^8 - 459512*x^7 + 846276*x^6 + 443584*x^5 - 527870*x^4 - 304834*x^3 + 104664*x^2 + 89954*x + 13729)
 
gp: K = bnfinit(x^20 - 6*x^19 - 38*x^18 + 282*x^17 + 379*x^16 - 4992*x^15 + 1570*x^14 + 41306*x^13 - 46263*x^12 - 166594*x^11 + 263516*x^10 + 353360*x^9 - 662559*x^8 - 459512*x^7 + 846276*x^6 + 443584*x^5 - 527870*x^4 - 304834*x^3 + 104664*x^2 + 89954*x + 13729, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 38 x^{18} + 282 x^{17} + 379 x^{16} - 4992 x^{15} + 1570 x^{14} + 41306 x^{13} - 46263 x^{12} - 166594 x^{11} + 263516 x^{10} + 353360 x^{9} - 662559 x^{8} - 459512 x^{7} + 846276 x^{6} + 443584 x^{5} - 527870 x^{4} - 304834 x^{3} + 104664 x^{2} + 89954 x + 13729 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7303816416639774784171798530359296=2^{30}\cdot 11^{16}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.34$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{54582450567325118917837803270583} a^{19} - \frac{7996504417711504336842960961534}{54582450567325118917837803270583} a^{18} + \frac{26699413164354314919287479621288}{54582450567325118917837803270583} a^{17} - \frac{18024174001253460406277350201203}{54582450567325118917837803270583} a^{16} - \frac{23235903048678227459721555561943}{54582450567325118917837803270583} a^{15} - \frac{13869962256033587823494372152975}{54582450567325118917837803270583} a^{14} - \frac{9269170216440021387814506547265}{54582450567325118917837803270583} a^{13} - \frac{17573817303234852261783261750593}{54582450567325118917837803270583} a^{12} - \frac{10743402829488055576903809247263}{54582450567325118917837803270583} a^{11} - \frac{2251993817584364262163566265645}{54582450567325118917837803270583} a^{10} - \frac{3358759417324316699956638077598}{54582450567325118917837803270583} a^{9} - \frac{26444375245970462366774751185102}{54582450567325118917837803270583} a^{8} + \frac{13146183167228658652753192172174}{54582450567325118917837803270583} a^{7} - \frac{7810339523408817912545931938483}{54582450567325118917837803270583} a^{6} + \frac{2331140132642050912083325325189}{54582450567325118917837803270583} a^{5} - \frac{17282155521378942444898995711530}{54582450567325118917837803270583} a^{4} - \frac{14796121463517177581974862689803}{54582450567325118917837803270583} a^{3} + \frac{17503613644188692441728900475333}{54582450567325118917837803270583} a^{2} - \frac{21667000789165222545592752204574}{54582450567325118917837803270583} a - \frac{27128177547754404011592624672715}{54582450567325118917837803270583}$

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:  $19$
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:  \( 41333671085.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T40):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 16 conjugacy class representatives for $C_2\times C_2^4:C_5$
Character table for $C_2\times C_2^4:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.2670699013250048.1, 10.10.5048580365312.1, 10.10.116117348402176.1

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 32 sibling: data not computed
Degree 40 siblings: data not computed
Arithmetically equvalently 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$