↑

Properties

Label 20.20.2108097156...7677.2
Degree $20$
Signature $[20, 0]$
Discriminant $11^{16}\cdot 43^{5}\cdot 199^{5}$
Root discriminant $65.49$
Ramified primes $11, 43, 199$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T310

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22307, -161926, 212326, 467149, -1034673, -241444, 1529604, -352662, -1021695, 466727, 334300, -214995, -51892, 48956, 2463, -5846, 257, 347, -33, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 33*x^18 + 347*x^17 + 257*x^16 - 5846*x^15 + 2463*x^14 + 48956*x^13 - 51892*x^12 - 214995*x^11 + 334300*x^10 + 466727*x^9 - 1021695*x^8 - 352662*x^7 + 1529604*x^6 - 241444*x^5 - 1034673*x^4 + 467149*x^3 + 212326*x^2 - 161926*x + 22307)
 
gp: K = bnfinit(x^20 - 8*x^19 - 33*x^18 + 347*x^17 + 257*x^16 - 5846*x^15 + 2463*x^14 + 48956*x^13 - 51892*x^12 - 214995*x^11 + 334300*x^10 + 466727*x^9 - 1021695*x^8 - 352662*x^7 + 1529604*x^6 - 241444*x^5 - 1034673*x^4 + 467149*x^3 + 212326*x^2 - 161926*x + 22307, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} - 33 x^{18} + 347 x^{17} + 257 x^{16} - 5846 x^{15} + 2463 x^{14} + 48956 x^{13} - 51892 x^{12} - 214995 x^{11} + 334300 x^{10} + 466727 x^{9} - 1021695 x^{8} - 352662 x^{7} + 1529604 x^{6} - 241444 x^{5} - 1034673 x^{4} + 467149 x^{3} + 212326 x^{2} - 161926 x + 22307 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2108097156584902878904709960659367677=11^{16}\cdot 43^{5}\cdot 199^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.49$
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:  $11, 43, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} - \frac{2}{11} a^{12} + \frac{3}{11} a^{11} + \frac{2}{11} a^{10} + \frac{3}{11} a^{9} + \frac{4}{11} a^{8} - \frac{2}{11} a^{7} + \frac{2}{11} a^{6} + \frac{1}{11} a^{5} + \frac{2}{11} a^{4} - \frac{2}{11} a^{2} + \frac{2}{11} a - \frac{3}{11}$, $\frac{1}{11} a^{14} - \frac{1}{11} a^{12} - \frac{3}{11} a^{11} - \frac{4}{11} a^{10} - \frac{1}{11} a^{9} - \frac{5}{11} a^{8} - \frac{2}{11} a^{7} + \frac{5}{11} a^{6} + \frac{4}{11} a^{5} + \frac{4}{11} a^{4} - \frac{2}{11} a^{3} - \frac{2}{11} a^{2} + \frac{1}{11} a + \frac{5}{11}$, $\frac{1}{11} a^{15} - \frac{5}{11} a^{12} - \frac{1}{11} a^{11} + \frac{1}{11} a^{10} - \frac{2}{11} a^{9} + \frac{2}{11} a^{8} + \frac{3}{11} a^{7} - \frac{5}{11} a^{6} + \frac{5}{11} a^{5} - \frac{2}{11} a^{3} - \frac{1}{11} a^{2} - \frac{4}{11} a - \frac{3}{11}$, $\frac{1}{11} a^{16} + \frac{5}{11} a^{11} - \frac{3}{11} a^{10} - \frac{5}{11} a^{9} + \frac{1}{11} a^{8} - \frac{4}{11} a^{7} + \frac{4}{11} a^{6} + \frac{5}{11} a^{5} - \frac{3}{11} a^{4} - \frac{1}{11} a^{3} - \frac{3}{11} a^{2} - \frac{4}{11} a - \frac{4}{11}$, $\frac{1}{11} a^{17} + \frac{5}{11} a^{12} - \frac{3}{11} a^{11} - \frac{5}{11} a^{10} + \frac{1}{11} a^{9} - \frac{4}{11} a^{8} + \frac{4}{11} a^{7} + \frac{5}{11} a^{6} - \frac{3}{11} a^{5} - \frac{1}{11} a^{4} - \frac{3}{11} a^{3} - \frac{4}{11} a^{2} - \frac{4}{11} a$, $\frac{1}{11} a^{18} - \frac{4}{11} a^{12} + \frac{2}{11} a^{11} + \frac{2}{11} a^{10} + \frac{3}{11} a^{9} - \frac{5}{11} a^{8} + \frac{4}{11} a^{7} - \frac{2}{11} a^{6} + \frac{5}{11} a^{5} - \frac{2}{11} a^{4} - \frac{4}{11} a^{3} - \frac{5}{11} a^{2} + \frac{1}{11} a + \frac{4}{11}$, $\frac{1}{160296562191323591332202877953760143} a^{19} - \frac{512155310364653093839979374097585}{14572414744665781030200261632160013} a^{18} + \frac{376712545355184485412362407295827}{14572414744665781030200261632160013} a^{17} - \frac{98242666101374941713833714716566}{160296562191323591332202877953760143} a^{16} - \frac{7149858927075410521144714955242064}{160296562191323591332202877953760143} a^{15} + \frac{1927125033452695180325157152789534}{160296562191323591332202877953760143} a^{14} + \frac{858628687592119083571199152628078}{160296562191323591332202877953760143} a^{13} + \frac{30625364907601479606288979191400386}{160296562191323591332202877953760143} a^{12} - \frac{74439211715895309360269957311041320}{160296562191323591332202877953760143} a^{11} - \frac{3493835354303508766038077881769454}{160296562191323591332202877953760143} a^{10} - \frac{31030250470332223844676507120782532}{160296562191323591332202877953760143} a^{9} - \frac{27155080044925589965053030546949411}{160296562191323591332202877953760143} a^{8} + \frac{42242841110826090822012976966835}{1223637879323080849864144106517253} a^{7} - \frac{10624138284382933447724376891344854}{160296562191323591332202877953760143} a^{6} - \frac{8354157581853420867474727032509605}{160296562191323591332202877953760143} a^{5} - \frac{64568112324736702801944251670849459}{160296562191323591332202877953760143} a^{4} + \frac{70435214385672235997387164389106424}{160296562191323591332202877953760143} a^{3} + \frac{69706764821475434707293257522005820}{160296562191323591332202877953760143} a^{2} - \frac{32399190891416509014545773059538427}{160296562191323591332202877953760143} a - \frac{9894563186347255179552980753867652}{160296562191323591332202877953760143}$

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:  $19$
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:  \( 334019448168 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T310:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 44 conjugacy class representatives for t20n310
Character table for t20n310 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.15695839359943369.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
43Data not computed
199Data not computed