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Properties

Label 20.20.1463368623...5625.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{20}\cdot 5^{10}\cdot 73^{10}$
Root discriminant $57.31$
Ramified primes $3, 5, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T277

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-31319, -62207, 263720, 631244, -257359, -1318055, -247384, 1127252, 468538, -490494, -275583, 116040, 82315, -14543, -13678, 806, 1256, -2, -58, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 58*x^18 - 2*x^17 + 1256*x^16 + 806*x^15 - 13678*x^14 - 14543*x^13 + 82315*x^12 + 116040*x^11 - 275583*x^10 - 490494*x^9 + 468538*x^8 + 1127252*x^7 - 247384*x^6 - 1318055*x^5 - 257359*x^4 + 631244*x^3 + 263720*x^2 - 62207*x - 31319)
 
gp: K = bnfinit(x^20 - x^19 - 58*x^18 - 2*x^17 + 1256*x^16 + 806*x^15 - 13678*x^14 - 14543*x^13 + 82315*x^12 + 116040*x^11 - 275583*x^10 - 490494*x^9 + 468538*x^8 + 1127252*x^7 - 247384*x^6 - 1318055*x^5 - 257359*x^4 + 631244*x^3 + 263720*x^2 - 62207*x - 31319, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 58 x^{18} - 2 x^{17} + 1256 x^{16} + 806 x^{15} - 13678 x^{14} - 14543 x^{13} + 82315 x^{12} + 116040 x^{11} - 275583 x^{10} - 490494 x^{9} + 468538 x^{8} + 1127252 x^{7} - 247384 x^{6} - 1318055 x^{5} - 257359 x^{4} + 631244 x^{3} + 263720 x^{2} - 62207 x - 31319 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(146336862347119602194701496572265625=3^{20}\cdot 5^{10}\cdot 73^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.31$
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:  $3, 5, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{14} - \frac{2}{5} a^{13} + \frac{2}{5} a^{11} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5}$, $\frac{1}{5} a^{17} + \frac{2}{5} a^{15} - \frac{2}{5} a^{14} + \frac{2}{5} a^{12} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a$, $\frac{1}{221545} a^{18} - \frac{5199}{221545} a^{17} - \frac{4037}{221545} a^{16} - \frac{10625}{44309} a^{15} + \frac{2875}{44309} a^{14} - \frac{9185}{44309} a^{13} - \frac{104978}{221545} a^{12} + \frac{41167}{221545} a^{11} + \frac{9221}{221545} a^{10} - \frac{81931}{221545} a^{9} - \frac{88826}{221545} a^{8} + \frac{34612}{221545} a^{7} - \frac{60169}{221545} a^{6} - \frac{81417}{221545} a^{5} - \frac{16221}{221545} a^{4} + \frac{39602}{221545} a^{3} + \frac{21184}{221545} a^{2} - \frac{109931}{221545} a - \frac{28981}{221545}$, $\frac{1}{1616813101296197129727567048595} a^{19} - \frac{1255138769878179253221144}{1616813101296197129727567048595} a^{18} - \frac{6486259946618138160249845797}{1616813101296197129727567048595} a^{17} - \frac{42156388827414410018255712078}{1616813101296197129727567048595} a^{16} - \frac{127451019851885711858792222099}{323362620259239425945513409719} a^{15} + \frac{142061918707308609982864080914}{1616813101296197129727567048595} a^{14} + \frac{582913754960314467577082722163}{1616813101296197129727567048595} a^{13} - \frac{503010800600904651759135719973}{1616813101296197129727567048595} a^{12} + \frac{66159096084307049433821462555}{323362620259239425945513409719} a^{11} + \frac{642967047212411239110504332529}{1616813101296197129727567048595} a^{10} - \frac{522531514629111733292833386296}{1616813101296197129727567048595} a^{9} + \frac{379142211797858222821117492529}{1616813101296197129727567048595} a^{8} + \frac{87683597280630281903927123717}{1616813101296197129727567048595} a^{7} - \frac{394620719699805458463259151792}{1616813101296197129727567048595} a^{6} - \frac{528162882390638572854213504433}{1616813101296197129727567048595} a^{5} - \frac{414591942662664740726621795354}{1616813101296197129727567048595} a^{4} - \frac{276017149855171437389776206382}{1616813101296197129727567048595} a^{3} + \frac{477885841883019923676201309424}{1616813101296197129727567048595} a^{2} + \frac{113981239576580615140692086004}{1616813101296197129727567048595} a - \frac{381658558749907513534946136047}{1616813101296197129727567048595}$

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:  $19$
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:  \( 187583748822 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T277:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 48 conjugacy class representatives for t20n277
Character table for t20n277 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.10791225.1, 10.10.382540014047053125.1, 10.10.582252685003125.1, 10.10.76508002809410625.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.12.14.6$x^{12} + 3 x^{11} + 3 x^{10} - 6 x^{9} + 3 x^{8} + 9 x^{7} + 9 x^{4} + 9 x^{3} + 9$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.12.10.3$x^{12} - 14527 x^{6} + 78021889$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$