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Properties

Label 20.0.16294596052...3125.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{8}\cdot 5^{17}\cdot 71^{10}$
Root discriminant $51.36$
Ramified primes $3, 5, 71$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $C_{10}^2:C_2^2$ (as 20T106)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![273375, -291600, 387585, -177390, 54681, 88536, -77348, 36503, 12058, -25026, 18597, -5839, -561, 2494, -1283, 456, 33, -53, 32, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 32*x^18 - 53*x^17 + 33*x^16 + 456*x^15 - 1283*x^14 + 2494*x^13 - 561*x^12 - 5839*x^11 + 18597*x^10 - 25026*x^9 + 12058*x^8 + 36503*x^7 - 77348*x^6 + 88536*x^5 + 54681*x^4 - 177390*x^3 + 387585*x^2 - 291600*x + 273375)
 
gp: K = bnfinit(x^20 - 6*x^19 + 32*x^18 - 53*x^17 + 33*x^16 + 456*x^15 - 1283*x^14 + 2494*x^13 - 561*x^12 - 5839*x^11 + 18597*x^10 - 25026*x^9 + 12058*x^8 + 36503*x^7 - 77348*x^6 + 88536*x^5 + 54681*x^4 - 177390*x^3 + 387585*x^2 - 291600*x + 273375, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 32 x^{18} - 53 x^{17} + 33 x^{16} + 456 x^{15} - 1283 x^{14} + 2494 x^{13} - 561 x^{12} - 5839 x^{11} + 18597 x^{10} - 25026 x^{9} + 12058 x^{8} + 36503 x^{7} - 77348 x^{6} + 88536 x^{5} + 54681 x^{4} - 177390 x^{3} + 387585 x^{2} - 291600 x + 273375 \)

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:  \(16294596052685417602356719970703125=3^{8}\cdot 5^{17}\cdot 71^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.36$
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:  $3, 5, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{2}{9} a^{10} + \frac{1}{9} a^{8} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} + \frac{1}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{18} a^{15} - \frac{1}{9} a^{12} - \frac{1}{6} a^{11} + \frac{1}{18} a^{10} + \frac{7}{18} a^{9} - \frac{1}{18} a^{8} + \frac{2}{9} a^{7} + \frac{1}{18} a^{6} - \frac{5}{18} a^{5} + \frac{1}{18} a^{4} + \frac{1}{9} a^{3} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{54} a^{16} + \frac{2}{27} a^{13} + \frac{1}{18} a^{12} - \frac{5}{54} a^{11} + \frac{7}{54} a^{10} + \frac{5}{54} a^{9} - \frac{7}{27} a^{8} + \frac{13}{54} a^{7} - \frac{5}{54} a^{6} + \frac{1}{54} a^{5} - \frac{11}{27} a^{4} - \frac{1}{3} a^{3} + \frac{2}{9} a^{2} - \frac{1}{6} a$, $\frac{1}{162} a^{17} + \frac{1}{162} a^{16} - \frac{4}{81} a^{14} - \frac{5}{162} a^{13} - \frac{7}{81} a^{12} - \frac{2}{81} a^{11} + \frac{13}{27} a^{10} + \frac{7}{18} a^{9} - \frac{13}{162} a^{8} - \frac{5}{81} a^{7} - \frac{35}{81} a^{6} - \frac{1}{2} a^{5} - \frac{20}{81} a^{4} + \frac{2}{9} a^{3} - \frac{23}{54} a^{2} - \frac{1}{6} a$, $\frac{1}{2430} a^{18} + \frac{2}{1215} a^{17} - \frac{1}{135} a^{16} - \frac{53}{2430} a^{15} + \frac{43}{2430} a^{14} + \frac{391}{2430} a^{13} + \frac{107}{2430} a^{12} - \frac{17}{135} a^{11} + \frac{83}{810} a^{10} - \frac{122}{1215} a^{9} - \frac{89}{1215} a^{8} - \frac{391}{2430} a^{7} - \frac{359}{810} a^{6} + \frac{443}{2430} a^{5} + \frac{43}{270} a^{4} + \frac{367}{810} a^{3} - \frac{1}{45} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{108405191908432752481076003669397420450} a^{19} + \frac{4634145156520073218996567191257239}{108405191908432752481076003669397420450} a^{18} + \frac{25171814076371156101511667895370393}{12045021323159194720119555963266380050} a^{17} + \frac{468054197338184874350229668142488981}{54202595954216376240538001834698710225} a^{16} + \frac{1738316710011681895813032984472963123}{108405191908432752481076003669397420450} a^{15} - \frac{2068953496462128870870672576114282517}{54202595954216376240538001834698710225} a^{14} - \frac{2456908284182989689601529181969659563}{108405191908432752481076003669397420450} a^{13} - \frac{346657258248638624989034413441723633}{4015007107719731573373185321088793350} a^{12} - \frac{2249348714066099714063648221419775376}{18067531984738792080179333944899570075} a^{11} + \frac{14174611047260223797663094884186150608}{54202595954216376240538001834698710225} a^{10} - \frac{17992363604247628150950465011124983279}{54202595954216376240538001834698710225} a^{9} - \frac{5948743016246014031905434850586471011}{108405191908432752481076003669397420450} a^{8} + \frac{7967042531054682689500369840967542298}{18067531984738792080179333944899570075} a^{7} + \frac{14277325370483548396658936399993849963}{108405191908432752481076003669397420450} a^{6} - \frac{251109173790584912252439293622324166}{6022510661579597360059777981633190025} a^{5} - \frac{6590794560889552087135463457921390983}{36135063969477584160358667889799140150} a^{4} + \frac{1522779271849978778321503636947255463}{4015007107719731573373185321088793350} a^{3} - \frac{90862520134522567145350211313892897}{803001421543946314674637064217758670} a^{2} - \frac{17555574462560732886311860298823298}{44611190085774795259702059123208815} a - \frac{16897165215889015422871575126372}{2974079339051653017313470608213921}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( 3039969466.12 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{10}^2:C_2^2$ (as 20T106):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 46 conjugacy class representatives for $C_{10}^2:C_2^2$
Character table for $C_{10}^2:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-71}) \), 4.0.25205.1, 10.0.57086944308984375.1

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ R R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ $20$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.10.12.10$x^{10} + 10 x^{8} + 20 x^{7} + 15 x^{6} - 5 x^{5} + 5 x^{4} + 5 x^{2} - 5 x + 7$$5$$2$$12$$D_{10}$$[3/2]_{2}^{2}$
$71$71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$