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Properties

Label 20.12.9761447374...3125.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{13}\cdot 97^{2}\cdot 419^{2}\cdot 695771^{2}$
Root discriminant $31.58$
Ramified primes $5, 97, 419, 695771$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1039

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -10, 607, 720, -1927, -1264, 1591, -1124, -186, 2003, -1008, -766, 1004, 117, -440, -30, 92, 13, -7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 7*x^18 + 13*x^17 + 92*x^16 - 30*x^15 - 440*x^14 + 117*x^13 + 1004*x^12 - 766*x^11 - 1008*x^10 + 2003*x^9 - 186*x^8 - 1124*x^7 + 1591*x^6 - 1264*x^5 - 1927*x^4 + 720*x^3 + 607*x^2 - 10*x - 1)
 
gp: K = bnfinit(x^20 - 4*x^19 - 7*x^18 + 13*x^17 + 92*x^16 - 30*x^15 - 440*x^14 + 117*x^13 + 1004*x^12 - 766*x^11 - 1008*x^10 + 2003*x^9 - 186*x^8 - 1124*x^7 + 1591*x^6 - 1264*x^5 - 1927*x^4 + 720*x^3 + 607*x^2 - 10*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 7 x^{18} + 13 x^{17} + 92 x^{16} - 30 x^{15} - 440 x^{14} + 117 x^{13} + 1004 x^{12} - 766 x^{11} - 1008 x^{10} + 2003 x^{9} - 186 x^{8} - 1124 x^{7} + 1591 x^{6} - 1264 x^{5} - 1927 x^{4} + 720 x^{3} + 607 x^{2} - 10 x - 1 \)

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:  \(976144737494384627696533203125=5^{13}\cdot 97^{2}\cdot 419^{2}\cdot 695771^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.58$
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:  $5, 97, 419, 695771$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11} a^{18} + \frac{1}{11} a^{17} + \frac{1}{11} a^{16} - \frac{1}{11} a^{15} + \frac{2}{11} a^{14} - \frac{1}{11} a^{13} + \frac{1}{11} a^{12} - \frac{2}{11} a^{11} - \frac{4}{11} a^{10} + \frac{3}{11} a^{8} + \frac{5}{11} a^{7} + \frac{2}{11} a^{6} + \frac{1}{11} a^{5} - \frac{4}{11} a^{4} - \frac{5}{11} a^{3} + \frac{5}{11} a^{2} + \frac{4}{11} a + \frac{4}{11}$, $\frac{1}{11657741507503343724398721620419} a^{19} - \frac{172602100111033047892177233268}{11657741507503343724398721620419} a^{18} + \frac{3841795955247946048602701800089}{11657741507503343724398721620419} a^{17} - \frac{4178308040182510568378166141929}{11657741507503343724398721620419} a^{16} - \frac{5411218565591422807112705547679}{11657741507503343724398721620419} a^{15} + \frac{5390913894180946787685239628601}{11657741507503343724398721620419} a^{14} + \frac{5218422772524215532248479377261}{11657741507503343724398721620419} a^{13} + \frac{5760599975791493682616785658213}{11657741507503343724398721620419} a^{12} - \frac{2792569122245070921560292169323}{11657741507503343724398721620419} a^{11} - \frac{5815098559841144876873245998284}{11657741507503343724398721620419} a^{10} - \frac{1001854577182329410064550755254}{11657741507503343724398721620419} a^{9} + \frac{1779821036218387245319302888091}{11657741507503343724398721620419} a^{8} - \frac{31054619841490729764096906002}{1059794682500303974945338329129} a^{7} + \frac{976384262225150612722473960681}{11657741507503343724398721620419} a^{6} - \frac{467755557853468466466275750702}{1059794682500303974945338329129} a^{5} - \frac{4886679587922103559237184240848}{11657741507503343724398721620419} a^{4} - \frac{3270024831386211228449159207039}{11657741507503343724398721620419} a^{3} - \frac{771551621545849031946805724811}{11657741507503343724398721620419} a^{2} - \frac{2830726333786601025320909346512}{11657741507503343724398721620419} a - \frac{4253362749099440689083819689302}{11657741507503343724398721620419}$

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:  \( 50948419.4327 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1039:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 378 conjugacy class representatives for t20n1039 are not computed
Character table for t20n1039 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.911025153125.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$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}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
97Data not computed
419Data not computed
695771Data not computed