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Properties

Label 20.6.67108864000...0000.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{48}\cdot 5^{22}$
Root discriminant $31.00$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4:S_5$ (as 20T120)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-229, 280, 1435, -1520, -2415, 2744, 1920, -2080, -535, 920, -383, -600, 325, 320, -110, -64, 25, 0, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 + 25*x^16 - 64*x^15 - 110*x^14 + 320*x^13 + 325*x^12 - 600*x^11 - 383*x^10 + 920*x^9 - 535*x^8 - 2080*x^7 + 1920*x^6 + 2744*x^5 - 2415*x^4 - 1520*x^3 + 1435*x^2 + 280*x - 229)
 
gp: K = bnfinit(x^20 - 5*x^18 + 25*x^16 - 64*x^15 - 110*x^14 + 320*x^13 + 325*x^12 - 600*x^11 - 383*x^10 + 920*x^9 - 535*x^8 - 2080*x^7 + 1920*x^6 + 2744*x^5 - 2415*x^4 - 1520*x^3 + 1435*x^2 + 280*x - 229, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{18} + 25 x^{16} - 64 x^{15} - 110 x^{14} + 320 x^{13} + 325 x^{12} - 600 x^{11} - 383 x^{10} + 920 x^{9} - 535 x^{8} - 2080 x^{7} + 1920 x^{6} + 2744 x^{5} - 2415 x^{4} - 1520 x^{3} + 1435 x^{2} + 280 x - 229 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-671088640000000000000000000000=-\,2^{48}\cdot 5^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.00$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{34} a^{18} + \frac{7}{34} a^{17} + \frac{3}{34} a^{16} + \frac{3}{17} a^{15} + \frac{6}{17} a^{14} + \frac{6}{17} a^{13} - \frac{4}{17} a^{12} + \frac{5}{17} a^{11} + \frac{9}{34} a^{10} + \frac{5}{34} a^{9} - \frac{3}{34} a^{8} + \frac{7}{17} a^{7} - \frac{4}{17} a^{6} + \frac{5}{17} a^{5} + \frac{3}{17} a^{4} - \frac{2}{17} a^{3} - \frac{3}{34} a^{2} - \frac{1}{2} a + \frac{11}{34}$, $\frac{1}{1760551616038202731568038457254} a^{19} + \frac{7062734511723562117611389425}{880275808019101365784019228627} a^{18} - \frac{53238966388458523084408885363}{1760551616038202731568038457254} a^{17} - \frac{11314036672869630034767588531}{51780929883476550928471719331} a^{16} + \frac{18026089042000587084762832582}{51780929883476550928471719331} a^{15} - \frac{232037977330421827979726059679}{880275808019101365784019228627} a^{14} + \frac{214185228494939241665496956493}{880275808019101365784019228627} a^{13} - \frac{340779180979529801977383229189}{880275808019101365784019228627} a^{12} + \frac{436968018896106588254616737209}{1760551616038202731568038457254} a^{11} + \frac{364481298942164238024503048686}{880275808019101365784019228627} a^{10} + \frac{466797131813486159244079756319}{1760551616038202731568038457254} a^{9} - \frac{190837422080923112490262863412}{880275808019101365784019228627} a^{8} + \frac{227765307806730492911581111539}{880275808019101365784019228627} a^{7} + \frac{185120951103382298498682264656}{880275808019101365784019228627} a^{6} - \frac{437906436675957787528787259687}{880275808019101365784019228627} a^{5} + \frac{29741454542602226331001206294}{880275808019101365784019228627} a^{4} - \frac{35709662970729392398392204971}{160050146912563884688003496114} a^{3} - \frac{259299269947154808267257719161}{880275808019101365784019228627} a^{2} - \frac{315577302319633628511867855001}{1760551616038202731568038457254} a + \frac{120280500827042981061960705743}{880275808019101365784019228627}$

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:  $12$
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:  \( 25992624.9925 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:S_5$ (as 20T120):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 480
The 19 conjugacy class representatives for $C_4:S_5$
Character table for $C_4:S_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.400.1, 5.3.3200000.1, 10.6.51200000000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: 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 $20$ R $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
5Data not computed