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Properties

Label 20.12.1348323859...4976.4
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 11^{16}\cdot 23^{4}$
Root discriminant $25.50$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T751

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, -249, -394, 465, 304, -1257, 458, 1443, -1288, -301, 1176, -497, -370, 379, -44, -100, 44, 6, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 6*x^18 + 44*x^17 - 100*x^16 - 44*x^15 + 379*x^14 - 370*x^13 - 497*x^12 + 1176*x^11 - 301*x^10 - 1288*x^9 + 1443*x^8 + 458*x^7 - 1257*x^6 + 304*x^5 + 465*x^4 - 394*x^3 - 249*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^20 - 6*x^19 + 6*x^18 + 44*x^17 - 100*x^16 - 44*x^15 + 379*x^14 - 370*x^13 - 497*x^12 + 1176*x^11 - 301*x^10 - 1288*x^9 + 1443*x^8 + 458*x^7 - 1257*x^6 + 304*x^5 + 465*x^4 - 394*x^3 - 249*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 6 x^{18} + 44 x^{17} - 100 x^{16} - 44 x^{15} + 379 x^{14} - 370 x^{13} - 497 x^{12} + 1176 x^{11} - 301 x^{10} - 1288 x^{9} + 1443 x^{8} + 458 x^{7} - 1257 x^{6} + 304 x^{5} + 465 x^{4} - 394 x^{3} - 249 x^{2} - 12 x + 1 \)

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:  \(13483238599952325260241534976=2^{20}\cdot 11^{16}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.50$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{563496260113489858979} a^{19} + \frac{69437704776540212154}{563496260113489858979} a^{18} + \frac{21492655751967331836}{563496260113489858979} a^{17} + \frac{158009797492309394768}{563496260113489858979} a^{16} - \frac{50939538322581483399}{563496260113489858979} a^{15} - \frac{192711564256839297217}{563496260113489858979} a^{14} + \frac{242588117514942324075}{563496260113489858979} a^{13} + \frac{96996763707136426083}{563496260113489858979} a^{12} - \frac{263607658392713658573}{563496260113489858979} a^{11} - \frac{227294813265986496091}{563496260113489858979} a^{10} + \frac{138722046409389298070}{563496260113489858979} a^{9} - \frac{225422904076654904779}{563496260113489858979} a^{8} - \frac{202373670680328021077}{563496260113489858979} a^{7} + \frac{231777514013842282798}{563496260113489858979} a^{6} + \frac{258466522645101261636}{563496260113489858979} a^{5} + \frac{221492472599541403344}{563496260113489858979} a^{4} + \frac{234404071334134103565}{563496260113489858979} a^{3} - \frac{278088895150550238212}{563496260113489858979} a^{2} - \frac{192818730545164370455}{563496260113489858979} a - \frac{226665309588457649185}{563496260113489858979}$

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:  \( 5967378.59862 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T751:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n751 are not computed
Character table for t20n751 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.219503494144.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$