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Properties

Label 20.0.51631781548...6576.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 37^{10}$
Root discriminant $17.20$
Ramified primes $2, 37$
Class number $1$
Class group Trivial
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![4, -16, 38, 48, -117, 10, 17, -10, 327, -788, 872, -524, 168, -40, 42, -48, 29, -10, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 3*x^18 - 10*x^17 + 29*x^16 - 48*x^15 + 42*x^14 - 40*x^13 + 168*x^12 - 524*x^11 + 872*x^10 - 788*x^9 + 327*x^8 - 10*x^7 + 17*x^6 + 10*x^5 - 117*x^4 + 48*x^3 + 38*x^2 - 16*x + 4)
 
gp: K = bnfinit(x^20 - 2*x^19 + 3*x^18 - 10*x^17 + 29*x^16 - 48*x^15 + 42*x^14 - 40*x^13 + 168*x^12 - 524*x^11 + 872*x^10 - 788*x^9 + 327*x^8 - 10*x^7 + 17*x^6 + 10*x^5 - 117*x^4 + 48*x^3 + 38*x^2 - 16*x + 4, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 3 x^{18} - 10 x^{17} + 29 x^{16} - 48 x^{15} + 42 x^{14} - 40 x^{13} + 168 x^{12} - 524 x^{11} + 872 x^{10} - 788 x^{9} + 327 x^{8} - 10 x^{7} + 17 x^{6} + 10 x^{5} - 117 x^{4} + 48 x^{3} + 38 x^{2} - 16 x + 4 \)

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:  \(5163178154897836475416576=2^{30}\cdot 37^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.20$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{17} - \frac{1}{8} a^{16} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{3}{8} a^{6} - \frac{1}{8} a^{5} - \frac{3}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{13018510997805166616} a^{19} + \frac{189295906102827221}{6509255498902583308} a^{18} + \frac{190747402299306533}{6509255498902583308} a^{17} - \frac{981979782376127861}{13018510997805166616} a^{16} - \frac{55170396274195387}{3254627749451291654} a^{15} + \frac{523399205975368941}{3254627749451291654} a^{14} + \frac{477859273278442541}{6509255498902583308} a^{13} - \frac{285957850748069527}{3254627749451291654} a^{12} - \frac{1407051139893203809}{6509255498902583308} a^{11} + \frac{777899288239329065}{6509255498902583308} a^{10} - \frac{620526032447326023}{3254627749451291654} a^{9} + \frac{197996555537946229}{929893642700369044} a^{8} + \frac{2825514860193642965}{13018510997805166616} a^{7} + \frac{1095450349833984085}{6509255498902583308} a^{6} + \frac{75456824844330049}{1627313874725645827} a^{5} - \frac{2959760360465002653}{13018510997805166616} a^{4} - \frac{162778964681753529}{464946821350184522} a^{3} - \frac{143370045430466335}{342592394679083332} a^{2} + \frac{16409806747348876}{85648098669770833} a + \frac{1442392527476815905}{3254627749451291654}$

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:  \( 31109.7391259 \)
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

\(\Q(\sqrt{-74}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{-2}, \sqrt{37})\), 5.1.87616.1 x5, 10.0.2272262782976.1, 10.0.61412507648.1 x5, 10.2.284032847872.1 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ 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$2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$37$37.4.2.1$x^{4} + 333 x^{2} + 34225$$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}$