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Properties

Label 20.0.25869668051...9761.1
Degree $20$
Signature $[0, 10]$
Discriminant $19^{12}\cdot 43^{8}$
Root discriminant $26.34$
Ramified primes $19, 43$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^4:D_5$ (as 20T73)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43, -215, 595, -545, -509, 1781, -1723, 2017, 63, -1671, 3380, -3625, 3373, -2358, 1489, -755, 333, -118, 34, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 34*x^18 - 118*x^17 + 333*x^16 - 755*x^15 + 1489*x^14 - 2358*x^13 + 3373*x^12 - 3625*x^11 + 3380*x^10 - 1671*x^9 + 63*x^8 + 2017*x^7 - 1723*x^6 + 1781*x^5 - 509*x^4 - 545*x^3 + 595*x^2 - 215*x + 43)
 
gp: K = bnfinit(x^20 - 7*x^19 + 34*x^18 - 118*x^17 + 333*x^16 - 755*x^15 + 1489*x^14 - 2358*x^13 + 3373*x^12 - 3625*x^11 + 3380*x^10 - 1671*x^9 + 63*x^8 + 2017*x^7 - 1723*x^6 + 1781*x^5 - 509*x^4 - 545*x^3 + 595*x^2 - 215*x + 43, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 34 x^{18} - 118 x^{17} + 333 x^{16} - 755 x^{15} + 1489 x^{14} - 2358 x^{13} + 3373 x^{12} - 3625 x^{11} + 3380 x^{10} - 1671 x^{9} + 63 x^{8} + 2017 x^{7} - 1723 x^{6} + 1781 x^{5} - 509 x^{4} - 545 x^{3} + 595 x^{2} - 215 x + 43 \)

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:  \(25869668051447537847885359761=19^{12}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.34$
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:  $19, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{17} + \frac{1}{5} a^{16} - \frac{1}{10} a^{15} - \frac{1}{10} a^{14} - \frac{2}{5} a^{13} - \frac{1}{2} a^{12} - \frac{1}{10} a^{11} + \frac{1}{10} a^{9} - \frac{3}{10} a^{8} - \frac{1}{2} a^{6} - \frac{3}{10} a^{5} - \frac{1}{2} a^{4} - \frac{2}{5} a^{3} - \frac{1}{10} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{70} a^{18} + \frac{1}{70} a^{17} - \frac{4}{35} a^{16} - \frac{3}{14} a^{15} + \frac{16}{35} a^{14} - \frac{1}{70} a^{13} - \frac{3}{10} a^{12} + \frac{8}{35} a^{11} + \frac{1}{70} a^{10} + \frac{3}{10} a^{9} - \frac{1}{35} a^{8} - \frac{1}{2} a^{7} - \frac{33}{70} a^{6} + \frac{3}{70} a^{5} + \frac{3}{10} a^{4} - \frac{1}{10} a^{3} + \frac{3}{7} a^{2} + \frac{17}{70} a + \frac{29}{70}$, $\frac{1}{13996707004856679972782852750} a^{19} + \frac{32232749999761821056591441}{6998353502428339986391426375} a^{18} + \frac{113477607531085478768643616}{6998353502428339986391426375} a^{17} - \frac{253246583069115037799399599}{2799341400971335994556570550} a^{16} + \frac{1463384487792801109991800403}{13996707004856679972782852750} a^{15} + \frac{51033337681033392359981758}{999764786061191426627346625} a^{14} - \frac{6061436230504300603099265643}{13996707004856679972782852750} a^{13} - \frac{1168173959569619852789265887}{2799341400971335994556570550} a^{12} - \frac{635406822763439906246970596}{6998353502428339986391426375} a^{11} - \frac{6087894873728143617719050863}{13996707004856679972782852750} a^{10} - \frac{2028313050737674895445344977}{13996707004856679972782852750} a^{9} + \frac{3462335012923092827673598388}{6998353502428339986391426375} a^{8} + \frac{1462017677199049095203715527}{13996707004856679972782852750} a^{7} + \frac{52914660499929291183129986}{199952957212238285325469325} a^{6} + \frac{4627716593442245252693148257}{13996707004856679972782852750} a^{5} + \frac{216141870820906343582396061}{999764786061191426627346625} a^{4} + \frac{1974956894404699713049648086}{6998353502428339986391426375} a^{3} - \frac{71631388855031432235677213}{285647081731768979036384750} a^{2} + \frac{196053957182384076794543}{3321477694555453244609125} a - \frac{42560159085086795822286392}{162752407033217208985847125}$

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:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 911154.170751 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T73):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

\(\Q(\sqrt{-19}) \), 5.5.667489.1, 10.0.8465289737299.1, 10.8.8465289737299.1, 10.2.160840505008681.1

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.8.6.2$x^{8} - 19 x^{4} + 5776$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$