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Properties

Label 20.12.2753714959...8125.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{6}\cdot 5^{15}\cdot 23^{4}\cdot 89^{7}$
Root discriminant $41.88$
Ramified primes $3, 5, 23, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T466

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -116, 143, 2849, -495, -10420, -3535, 8747, 6436, 102, -4694, -1338, 1636, -552, 197, 208, -141, 11, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 3*x^18 + 11*x^17 - 141*x^16 + 208*x^15 + 197*x^14 - 552*x^13 + 1636*x^12 - 1338*x^11 - 4694*x^10 + 102*x^9 + 6436*x^8 + 8747*x^7 - 3535*x^6 - 10420*x^5 - 495*x^4 + 2849*x^3 + 143*x^2 - 116*x - 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 3*x^18 + 11*x^17 - 141*x^16 + 208*x^15 + 197*x^14 - 552*x^13 + 1636*x^12 - 1338*x^11 - 4694*x^10 + 102*x^9 + 6436*x^8 + 8747*x^7 - 3535*x^6 - 10420*x^5 - 495*x^4 + 2849*x^3 + 143*x^2 - 116*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 3 x^{18} + 11 x^{17} - 141 x^{16} + 208 x^{15} + 197 x^{14} - 552 x^{13} + 1636 x^{12} - 1338 x^{11} - 4694 x^{10} + 102 x^{9} + 6436 x^{8} + 8747 x^{7} - 3535 x^{6} - 10420 x^{5} - 495 x^{4} + 2849 x^{3} + 143 x^{2} - 116 x - 1 \)

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:  \(275371495990487787417022705078125=3^{6}\cdot 5^{15}\cdot 23^{4}\cdot 89^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.88$
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:  $3, 5, 23, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{16} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{16} - \frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{906773469379745743417745633256345759} a^{19} + \frac{29765775386011732632136826075268219}{302257823126581914472581877752115253} a^{18} - \frac{39221950786386577559759038011584114}{906773469379745743417745633256345759} a^{17} + \frac{148483848273760591396283553404981317}{302257823126581914472581877752115253} a^{16} - \frac{15182349803991677504702236508085418}{302257823126581914472581877752115253} a^{15} - \frac{147606732747581431852406194167295890}{302257823126581914472581877752115253} a^{14} + \frac{94439669885722112012918210591510083}{302257823126581914472581877752115253} a^{13} + \frac{73628034728193843513536751496730998}{906773469379745743417745633256345759} a^{12} - \frac{137711946219802939872786246409194608}{302257823126581914472581877752115253} a^{11} + \frac{360505450414058191795363342594693670}{906773469379745743417745633256345759} a^{10} + \frac{111022683025307278122067337023008737}{302257823126581914472581877752115253} a^{9} + \frac{52172917763042609813054044151965007}{302257823126581914472581877752115253} a^{8} + \frac{452253985212473002720956458802126206}{906773469379745743417745633256345759} a^{7} + \frac{407508491968614015794308407308073260}{906773469379745743417745633256345759} a^{6} - \frac{384082669963320800041013204778282254}{906773469379745743417745633256345759} a^{5} + \frac{288317329380932918010406835126171500}{906773469379745743417745633256345759} a^{4} + \frac{379711874015523324872765407821026467}{906773469379745743417745633256345759} a^{3} - \frac{145065625838680792325097008360736291}{302257823126581914472581877752115253} a^{2} + \frac{4698589107337305427861609986383229}{302257823126581914472581877752115253} a - \frac{355048070933831705513202381545060272}{906773469379745743417745633256345759}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 541329393.385 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.767625.1, 10.10.2946240703125.1

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 $20$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
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$$[\ ]_{4}^{2}$
5Data not computed
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89Data not computed