Properties

Label 20.0.282...373.1
Degree $20$
Signature $[0, 10]$
Discriminant $2.826\times 10^{21}$
Root discriminant $11.82$
Ramified primes $3, 37, 109, 241$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group 20T781

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 24*x^18 - 52*x^17 + 75*x^16 - 58*x^15 - 28*x^14 + 140*x^13 - 109*x^12 - 274*x^11 + 1033*x^10 - 1874*x^9 + 2367*x^8 - 2282*x^7 + 1739*x^6 - 1057*x^5 + 509*x^4 - 190*x^3 + 53*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^20 - 7*x^19 + 24*x^18 - 52*x^17 + 75*x^16 - 58*x^15 - 28*x^14 + 140*x^13 - 109*x^12 - 274*x^11 + 1033*x^10 - 1874*x^9 + 2367*x^8 - 2282*x^7 + 1739*x^6 - 1057*x^5 + 509*x^4 - 190*x^3 + 53*x^2 - 10*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 53, -190, 509, -1057, 1739, -2282, 2367, -1874, 1033, -274, -109, 140, -28, -58, 75, -52, 24, -7, 1]);
 

\( x^{20} - 7 x^{19} + 24 x^{18} - 52 x^{17} + 75 x^{16} - 58 x^{15} - 28 x^{14} + 140 x^{13} - 109 x^{12} - 274 x^{11} + 1033 x^{10} - 1874 x^{9} + 2367 x^{8} - 2282 x^{7} + 1739 x^{6} - 1057 x^{5} + 509 x^{4} - 190 x^{3} + 53 x^{2} - 10 x + 1 \)

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(2825584168318748978373\)\(\medspace = 3^{10}\cdot 37^{5}\cdot 109^{2}\cdot 241^{2}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $11.82$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 37, 109, 241$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{58974943} a^{19} + \frac{26100956}{58974943} a^{18} + \frac{4441811}{58974943} a^{17} + \frac{6717049}{58974943} a^{16} + \frac{29168546}{58974943} a^{15} + \frac{21041664}{58974943} a^{14} - \frac{1661676}{58974943} a^{13} + \frac{8787212}{58974943} a^{12} + \frac{21434130}{58974943} a^{11} + \frac{27183508}{58974943} a^{10} - \frac{7725163}{58974943} a^{9} + \frac{15959240}{58974943} a^{8} + \frac{10227602}{58974943} a^{7} + \frac{4964115}{58974943} a^{6} - \frac{3726288}{58974943} a^{5} + \frac{1530909}{58974943} a^{4} + \frac{21410941}{58974943} a^{3} - \frac{27182349}{58974943} a^{2} + \frac{4596607}{58974943} a + \frac{17015652}{58974943}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{603522}{140083} a^{19} - \frac{4064809}{140083} a^{18} + \frac{13277554}{140083} a^{17} - \frac{27061215}{140083} a^{16} + \frac{35690210}{140083} a^{15} - \frac{21127454}{140083} a^{14} - \frac{27499899}{140083} a^{13} + \frac{78545176}{140083} a^{12} - \frac{37794123}{140083} a^{11} - \frac{188496067}{140083} a^{10} + \frac{573679398}{140083} a^{9} - \frac{936961751}{140083} a^{8} + \frac{1079862509}{140083} a^{7} - \frac{948908718}{140083} a^{6} + \frac{654085764}{140083} a^{5} - \frac{354326695}{140083} a^{4} + \frac{148436724}{140083} a^{3} - \frac{46655711}{140083} a^{2} + \frac{10373793}{140083} a - \frac{1206961}{140083} \) (order $6$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 567.746569822 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{10}\cdot 567.746569822 \cdot 1}{6\sqrt{2825584168318748978373}}\approx 0.170705623401$ (assuming GRH)

Galois group

20T781:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 115200
The 119 conjugacy class representatives for t20n781 are not computed
Character table for t20n781 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.333.1, 10.0.236184579.1

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 24 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 $20$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ $20$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
37Data not computed
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$
241Data not computed