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Properties

Label 20.8.92233720368...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{75}\cdot 5^{12}$
Root discriminant $35.34$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T654

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, -256, -752, 0, -4, 480, 832, -128, 376, -352, -608, 224, -164, 16, 192, -32, 2, 0, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 12*x^18 + 2*x^16 - 32*x^15 + 192*x^14 + 16*x^13 - 164*x^12 + 224*x^11 - 608*x^10 - 352*x^9 + 376*x^8 - 128*x^7 + 832*x^6 + 480*x^5 - 4*x^4 - 752*x^2 - 256*x - 8)
 
gp: K = bnfinit(x^20 - 12*x^18 + 2*x^16 - 32*x^15 + 192*x^14 + 16*x^13 - 164*x^12 + 224*x^11 - 608*x^10 - 352*x^9 + 376*x^8 - 128*x^7 + 832*x^6 + 480*x^5 - 4*x^4 - 752*x^2 - 256*x - 8, 1)
 

Normalized defining polynomial

\( x^{20} - 12 x^{18} + 2 x^{16} - 32 x^{15} + 192 x^{14} + 16 x^{13} - 164 x^{12} + 224 x^{11} - 608 x^{10} - 352 x^{9} + 376 x^{8} - 128 x^{7} + 832 x^{6} + 480 x^{5} - 4 x^{4} - 752 x^{2} - 256 x - 8 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9223372036854775808000000000000=2^{75}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.34$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{16}$, $\frac{1}{8} a^{17} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{248} a^{18} + \frac{1}{62} a^{17} - \frac{3}{124} a^{16} - \frac{1}{124} a^{15} - \frac{1}{31} a^{14} + \frac{9}{124} a^{13} - \frac{7}{62} a^{12} - \frac{7}{62} a^{11} - \frac{29}{124} a^{10} + \frac{3}{31} a^{9} - \frac{7}{31} a^{8} + \frac{13}{62} a^{7} + \frac{7}{62} a^{6} + \frac{21}{62} a^{5} - \frac{13}{31} a^{4} - \frac{4}{31} a^{3} + \frac{5}{31} a^{2} - \frac{2}{31} a + \frac{3}{31}$, $\frac{1}{74719404606866736248} a^{19} + \frac{16083383374658357}{74719404606866736248} a^{18} - \frac{1697182405312060055}{37359702303433368124} a^{17} + \frac{2043830702909075475}{18679851151716684062} a^{16} + \frac{433429825776616315}{18679851151716684062} a^{15} + \frac{2228605183524436951}{37359702303433368124} a^{14} + \frac{1510707095378863503}{18679851151716684062} a^{13} - \frac{1986010243653319855}{18679851151716684062} a^{12} - \frac{115440606725642787}{37359702303433368124} a^{11} + \frac{8418422833873387921}{37359702303433368124} a^{10} - \frac{118739667076543170}{1334275082265477433} a^{9} + \frac{1523800574527083375}{9339925575858342031} a^{8} - \frac{5928744225806337171}{18679851151716684062} a^{7} + \frac{509805530238982451}{1334275082265477433} a^{6} - \frac{20825860731145961}{9339925575858342031} a^{5} - \frac{957653288000935062}{9339925575858342031} a^{4} + \frac{251633197742527186}{1334275082265477433} a^{3} + \frac{568331523973135620}{1334275082265477433} a^{2} + \frac{596265177158700249}{9339925575858342031} a + \frac{3352726666744107853}{9339925575858342031}$

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:  $13$
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:  \( 329987043.074 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T654:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 70 conjugacy class representatives for t20n654 are not computed
Character table for t20n654 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 10.4.67108864000000.1

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 R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ $20$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
2Data not computed
5Data not computed