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Properties

Label 20.8.17165941082...8576.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{40}\cdot 11^{16}\cdot 23^{8}\cdot 199^{2}\cdot 331^{2}$
Root discriminant $289.56$
Ramified primes $2, 11, 23, 199, 331$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group 20T331

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![277717458830449, 0, 701568715396758, 0, 450513883311701, 0, 8416391205200, 0, -1371610846859, 0, 6739768564, 0, 566715315, 0, -3480624, 0, -54002, 0, 80, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 80*x^18 - 54002*x^16 - 3480624*x^14 + 566715315*x^12 + 6739768564*x^10 - 1371610846859*x^8 + 8416391205200*x^6 + 450513883311701*x^4 + 701568715396758*x^2 + 277717458830449)
 
gp: K = bnfinit(x^20 + 80*x^18 - 54002*x^16 - 3480624*x^14 + 566715315*x^12 + 6739768564*x^10 - 1371610846859*x^8 + 8416391205200*x^6 + 450513883311701*x^4 + 701568715396758*x^2 + 277717458830449, 1)
 

Normalized defining polynomial

\( x^{20} + 80 x^{18} - 54002 x^{16} - 3480624 x^{14} + 566715315 x^{12} + 6739768564 x^{10} - 1371610846859 x^{8} + 8416391205200 x^{6} + 450513883311701 x^{4} + 701568715396758 x^{2} + 277717458830449 \)

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:  \(17165941082918625907388507845862234378814751768576=2^{40}\cdot 11^{16}\cdot 23^{8}\cdot 199^{2}\cdot 331^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $289.56$
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:  $2, 11, 23, 199, 331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{23} a^{14} + \frac{9}{23} a^{12} + \frac{7}{23} a^{10} - \frac{2}{23} a^{8} + \frac{11}{23} a^{6} - \frac{3}{23} a^{4}$, $\frac{1}{253} a^{15} + \frac{124}{253} a^{13} + \frac{30}{253} a^{11} - \frac{48}{253} a^{9} + \frac{34}{253} a^{7} + \frac{89}{253} a^{5}$, $\frac{1}{253} a^{16} + \frac{3}{253} a^{14} - \frac{47}{253} a^{12} + \frac{117}{253} a^{10} + \frac{1}{11} a^{8} + \frac{1}{11} a^{6} + \frac{10}{23} a^{4}$, $\frac{1}{253} a^{17} + \frac{87}{253} a^{13} + \frac{27}{253} a^{11} - \frac{86}{253} a^{9} - \frac{79}{253} a^{7} + \frac{96}{253} a^{5}$, $\frac{1}{51645428878770915455720534168609932689149729024433817949955197257496163613} a^{18} - \frac{7419772555905286559865436251337560356588313636045142926114777918251434}{4695038988979174132338230378964539335377248093130347086359563387045105783} a^{16} - \frac{528913000626916468883712589797568901191805417572429925387246786404145473}{51645428878770915455720534168609932689149729024433817949955197257496163613} a^{14} - \frac{2837254730468324801459484807003689434161018846533839200203132360679716296}{51645428878770915455720534168609932689149729024433817949955197257496163613} a^{12} + \frac{11906696932203707896795029432346423339925198159224387809954333408094833583}{51645428878770915455720534168609932689149729024433817949955197257496163613} a^{10} - \frac{22983633589614324138213176702252502925335497030667586275398122041699758168}{51645428878770915455720534168609932689149729024433817949955197257496163613} a^{8} - \frac{1586152143470669076304855260763592974097920326363319817528982711710573096}{51645428878770915455720534168609932689149729024433817949955197257496163613} a^{6} - \frac{1234603077790767906685648862322991194019938673097645918509643397862073573}{4695038988979174132338230378964539335377248093130347086359563387045105783} a^{4} + \frac{30601411837801823561276734079378249806869247164123158883565842292624860}{204132129955616266623401320824545188494662960570884655928676669001961121} a^{2} + \frac{44810674758652110816553405945354395791957948397861118874265281097}{3099062228903069222599421895345992629228665389954070289949394540709}$, $\frac{1}{51645428878770915455720534168609932689149729024433817949955197257496163613} a^{19} - \frac{7419772555905286559865436251337560356588313636045142926114777918251434}{4695038988979174132338230378964539335377248093130347086359563387045105783} a^{17} + \frac{83483389239932330986491372676066664292183464140224042398783220601737890}{51645428878770915455720534168609932689149729024433817949955197257496163613} a^{15} + \frac{21454468734250010926725272371117187996703873461401434855309391250553657103}{51645428878770915455720534168609932689149729024433817949955197257496163613} a^{13} - \frac{1942440022778340323892671442004949307700442219439073736947269421747711740}{4695038988979174132338230378964539335377248093130347086359563387045105783} a^{11} - \frac{733231424452151076262432732377077379417234328441158779172365120485995979}{51645428878770915455720534168609932689149729024433817949955197257496163613} a^{9} + \frac{19235325112002190119282079463340016252357701651866915087196037526489461246}{51645428878770915455720534168609932689149729024433817949955197257496163613} a^{7} - \frac{10722784036319819240814518994009270495294043956081719920604604010455353609}{51645428878770915455720534168609932689149729024433817949955197257496163613} a^{5} + \frac{30601411837801823561276734079378249806869247164123158883565842292624860}{204132129955616266623401320824545188494662960570884655928676669001961121} a^{3} + \frac{44810674758652110816553405945354395791957948397861118874265281097}{3099062228903069222599421895345992629228665389954070289949394540709} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( 14609277926100000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T331:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 56 conjugacy class representatives for t20n331 are not computed
Character table for t20n331 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.2670699013250048.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$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}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$199$$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.4.2.2$x^{4} - 199 x^{2} + 237606$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
199.4.0.1$x^{4} - x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
331Data not computed