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Properties

Label 20.0.52870944306...3824.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 37^{10}$
Root discriminant $24.33$
Ramified primes $2, 37$
Class number $2$
Class group $[2]$
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -32, 544, -2544, 3012, 1408, -520, -8648, 7572, 56, -48, -176, 645, -796, 402, -132, 30, -12, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 10*x^18 - 12*x^17 + 30*x^16 - 132*x^15 + 402*x^14 - 796*x^13 + 645*x^12 - 176*x^11 - 48*x^10 + 56*x^9 + 7572*x^8 - 8648*x^7 - 520*x^6 + 1408*x^5 + 3012*x^4 - 2544*x^3 + 544*x^2 - 32*x + 16)
 
gp: K = bnfinit(x^20 - 4*x^19 + 10*x^18 - 12*x^17 + 30*x^16 - 132*x^15 + 402*x^14 - 796*x^13 + 645*x^12 - 176*x^11 - 48*x^10 + 56*x^9 + 7572*x^8 - 8648*x^7 - 520*x^6 + 1408*x^5 + 3012*x^4 - 2544*x^3 + 544*x^2 - 32*x + 16, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 10 x^{18} - 12 x^{17} + 30 x^{16} - 132 x^{15} + 402 x^{14} - 796 x^{13} + 645 x^{12} - 176 x^{11} - 48 x^{10} + 56 x^{9} + 7572 x^{8} - 8648 x^{7} - 520 x^{6} + 1408 x^{5} + 3012 x^{4} - 2544 x^{3} + 544 x^{2} - 32 x + 16 \)

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:  \(5287094430615384550826573824=2^{40}\cdot 37^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.33$
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:  $2, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{11} + \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{3}{8} a^{7} + \frac{3}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{14} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} + \frac{3}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{15} - \frac{1}{4} a^{11} + \frac{3}{8} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{136} a^{16} - \frac{1}{34} a^{15} - \frac{1}{136} a^{14} + \frac{1}{34} a^{13} - \frac{7}{68} a^{12} + \frac{5}{68} a^{11} - \frac{9}{68} a^{10} - \frac{13}{68} a^{9} + \frac{11}{136} a^{8} + \frac{33}{68} a^{7} + \frac{29}{136} a^{6} - \frac{1}{68} a^{5} - \frac{1}{68} a^{4} - \frac{15}{34} a^{3} - \frac{33}{68} a^{2} + \frac{11}{34} a - \frac{7}{34}$, $\frac{1}{136} a^{17} + \frac{1}{68} a^{13} - \frac{3}{34} a^{12} - \frac{3}{34} a^{11} + \frac{1}{34} a^{10} - \frac{25}{136} a^{9} + \frac{1}{17} a^{8} - \frac{8}{17} a^{7} + \frac{3}{34} a^{6} - \frac{5}{68} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{2}{17} a^{2} - \frac{7}{17} a + \frac{3}{17}$, $\frac{1}{54128} a^{18} + \frac{35}{27064} a^{17} + \frac{23}{6766} a^{16} + \frac{941}{27064} a^{15} - \frac{465}{27064} a^{14} - \frac{481}{13532} a^{13} - \frac{1051}{27064} a^{12} + \frac{151}{6766} a^{11} - \frac{13155}{54128} a^{10} + \frac{5883}{27064} a^{9} - \frac{4843}{27064} a^{8} - \frac{5019}{27064} a^{7} - \frac{3771}{13532} a^{6} - \frac{3355}{6766} a^{5} + \frac{130}{3383} a^{4} - \frac{3699}{13532} a^{3} + \frac{1447}{13532} a^{2} - \frac{1089}{3383} a + \frac{841}{6766}$, $\frac{1}{338111133295736990425572112} a^{19} + \frac{1793200809403236015885}{338111133295736990425572112} a^{18} - \frac{33380576476259375082071}{24150795235409785030398008} a^{17} + \frac{302990405517249914283005}{84527783323934247606393028} a^{16} - \frac{1871386789667884965066625}{84527783323934247606393028} a^{15} - \frac{567693488669438007489379}{84527783323934247606393028} a^{14} + \frac{2465089956941485781076817}{169055566647868495212786056} a^{13} + \frac{1728897931918146533297815}{169055566647868495212786056} a^{12} - \frac{27661811879343120934860947}{338111133295736990425572112} a^{11} - \frac{43678712045292394469661791}{338111133295736990425572112} a^{10} + \frac{10421334290905117609882213}{42263891661967123803196514} a^{9} - \frac{7447976294650698917852971}{169055566647868495212786056} a^{8} - \frac{23909509203110961594588507}{169055566647868495212786056} a^{7} + \frac{20471181186990240994582865}{169055566647868495212786056} a^{6} - \frac{37073080837665675137131985}{84527783323934247606393028} a^{5} - \frac{1460052088934146955550613}{42263891661967123803196514} a^{4} + \frac{2270915865437085808393881}{6037698808852446257599502} a^{3} - \frac{12982593874689605146202743}{42263891661967123803196514} a^{2} + \frac{6446025743774904499112580}{21131945830983561901598257} a - \frac{5021904995954097869821386}{21131945830983561901598257}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( 3559558.36635 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-74}) \), \(\Q(\sqrt{-37}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{-37})\), 5.1.87616.1 x5, 10.0.2272262782976.1, 10.0.18178102263808.2 x5, 10.2.982600122368.1 x5

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
$37$37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$