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Properties

Label 20.12.1816897514...6169.1
Degree $20$
Signature $[12, 4]$
Discriminant $13^{2}\cdot 401^{10}$
Root discriminant $25.88$
Ramified primes $13, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T81)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -40, -32, 366, 25, -1274, 563, 1876, -1358, -1215, 841, 941, -678, -177, 118, 99, -58, -14, 22, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 22*x^18 - 14*x^17 - 58*x^16 + 99*x^15 + 118*x^14 - 177*x^13 - 678*x^12 + 941*x^11 + 841*x^10 - 1215*x^9 - 1358*x^8 + 1876*x^7 + 563*x^6 - 1274*x^5 + 25*x^4 + 366*x^3 - 32*x^2 - 40*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 22*x^18 - 14*x^17 - 58*x^16 + 99*x^15 + 118*x^14 - 177*x^13 - 678*x^12 + 941*x^11 + 841*x^10 - 1215*x^9 - 1358*x^8 + 1876*x^7 + 563*x^6 - 1274*x^5 + 25*x^4 + 366*x^3 - 32*x^2 - 40*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 22 x^{18} - 14 x^{17} - 58 x^{16} + 99 x^{15} + 118 x^{14} - 177 x^{13} - 678 x^{12} + 941 x^{11} + 841 x^{10} - 1215 x^{9} - 1358 x^{8} + 1876 x^{7} + 563 x^{6} - 1274 x^{5} + 25 x^{4} + 366 x^{3} - 32 x^{2} - 40 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(18168975145356050963137476169=13^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.88$
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:  $13, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13} a^{18} + \frac{3}{13} a^{17} - \frac{5}{13} a^{16} - \frac{2}{13} a^{15} - \frac{1}{13} a^{14} - \frac{4}{13} a^{12} + \frac{4}{13} a^{10} - \frac{3}{13} a^{9} - \frac{4}{13} a^{8} + \frac{4}{13} a^{5} - \frac{4}{13} a^{4} + \frac{2}{13} a^{3} + \frac{1}{13} a^{2} - \frac{3}{13} a + \frac{5}{13}$, $\frac{1}{396591769297560379231} a^{19} + \frac{13028722776428538734}{396591769297560379231} a^{18} - \frac{12833989806548872508}{30507059176735413787} a^{17} + \frac{58108676325725633953}{396591769297560379231} a^{16} + \frac{4206288462039902978}{30507059176735413787} a^{15} - \frac{121252951056310740543}{396591769297560379231} a^{14} - \frac{25732439992680342705}{396591769297560379231} a^{13} + \frac{103297950802051300921}{396591769297560379231} a^{12} + \frac{48890439150531136045}{396591769297560379231} a^{11} - \frac{94075676595930078912}{396591769297560379231} a^{10} + \frac{103582855723058033313}{396591769297560379231} a^{9} + \frac{191930773170927365122}{396591769297560379231} a^{8} - \frac{13444991071030306441}{30507059176735413787} a^{7} + \frac{126713500628783305397}{396591769297560379231} a^{6} - \frac{114364021083208598643}{396591769297560379231} a^{5} + \frac{149729663198673607053}{396591769297560379231} a^{4} - \frac{2625624711321845485}{30507059176735413787} a^{3} + \frac{83774272695543698479}{396591769297560379231} a^{2} + \frac{10685631198850816752}{30507059176735413787} a + \frac{44204863820262022284}{396591769297560379231}$

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:  $15$
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:  \( 7695357.26661 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T81):

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

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.6.134792340826013.1, 10.6.336140500813.1, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed