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Properties

Label 20.0.10800753337...5625.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 31^{4}\cdot 71^{4}\cdot 2161^{5}$
Root discriminant $71.07$
Ramified primes $5, 31, 71, 2161$
Class number $11$ (GRH)
Class group $[11]$ (GRH)
Galois group 20T168

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![50483698079, -25512798670, 48965425278, -21172817140, 20977732177, -7728153702, 5194846005, -1623921679, 820377495, -215819750, 86065041, -18764856, 6059682, -1066070, 282210, -38164, 8304, -782, 139, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 139*x^18 - 782*x^17 + 8304*x^16 - 38164*x^15 + 282210*x^14 - 1066070*x^13 + 6059682*x^12 - 18764856*x^11 + 86065041*x^10 - 215819750*x^9 + 820377495*x^8 - 1623921679*x^7 + 5194846005*x^6 - 7728153702*x^5 + 20977732177*x^4 - 21172817140*x^3 + 48965425278*x^2 - 25512798670*x + 50483698079)
 
gp: K = bnfinit(x^20 - 7*x^19 + 139*x^18 - 782*x^17 + 8304*x^16 - 38164*x^15 + 282210*x^14 - 1066070*x^13 + 6059682*x^12 - 18764856*x^11 + 86065041*x^10 - 215819750*x^9 + 820377495*x^8 - 1623921679*x^7 + 5194846005*x^6 - 7728153702*x^5 + 20977732177*x^4 - 21172817140*x^3 + 48965425278*x^2 - 25512798670*x + 50483698079, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 139 x^{18} - 782 x^{17} + 8304 x^{16} - 38164 x^{15} + 282210 x^{14} - 1066070 x^{13} + 6059682 x^{12} - 18764856 x^{11} + 86065041 x^{10} - 215819750 x^{9} + 820377495 x^{8} - 1623921679 x^{7} + 5194846005 x^{6} - 7728153702 x^{5} + 20977732177 x^{4} - 21172817140 x^{3} + 48965425278 x^{2} - 25512798670 x + 50483698079 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10800753337779736826690692535166015625=5^{10}\cdot 31^{4}\cdot 71^{4}\cdot 2161^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.07$
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:  $5, 31, 71, 2161$
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}{1892766608947720396511197522287844567851851523463485943083110481401244165531} a^{19} - \frac{94644324984671088463345102568184081378799251300546290804983657062368855002}{630922202982573465503732507429281522617283841154495314361036827133748055177} a^{18} + \frac{3478902472205682376894983567704913420633236823246609213411443907243289090}{6984378630803396297089289750139647851851850640086663996616643842809019061} a^{17} - \frac{524865528462330816265968990751744269152444906791738268225085534159094906582}{1892766608947720396511197522287844567851851523463485943083110481401244165531} a^{16} + \frac{162799904174405839810552222602070557838109347329871950851453310745904275349}{1892766608947720396511197522287844567851851523463485943083110481401244165531} a^{15} + \frac{621121064425392564742717717670333291611814511415219164460147515006591650707}{1892766608947720396511197522287844567851851523463485943083110481401244165531} a^{14} - \frac{43462120713929470951904346514877170671360925635506437135392146946740076356}{1892766608947720396511197522287844567851851523463485943083110481401244165531} a^{13} - \frac{790942309601741514215858966949742898609374330290712563026792025174098724538}{1892766608947720396511197522287844567851851523463485943083110481401244165531} a^{12} - \frac{746833223675828386492624952908572018992473244628684386042020902000634101007}{1892766608947720396511197522287844567851851523463485943083110481401244165531} a^{11} - \frac{806622134395234173987521444253464326695360175002933826587295293678716378668}{1892766608947720396511197522287844567851851523463485943083110481401244165531} a^{10} + \frac{874330426356735051146701977746111289311618733311394608347802485402193568062}{1892766608947720396511197522287844567851851523463485943083110481401244165531} a^{9} + \frac{311283221863579238995289580652653843451867602709007941477972498119827847909}{630922202982573465503732507429281522617283841154495314361036827133748055177} a^{8} + \frac{169376776938193625274971991787358264678563962354310714133134004389772627887}{630922202982573465503732507429281522617283841154495314361036827133748055177} a^{7} - \frac{33475204728781827249027147034703678747335939635588811649988682062466063828}{99619295207774757711115659067781293044834290708604523320163709547433903449} a^{6} + \frac{748701145542608206369257850375031125407339621885535456552313194537018594730}{1892766608947720396511197522287844567851851523463485943083110481401244165531} a^{5} + \frac{353667835144592735240417324882036740642450174901079962252963982099026511188}{1892766608947720396511197522287844567851851523463485943083110481401244165531} a^{4} + \frac{75694740682136233620908109069974866987961416573340162971747322453241979231}{630922202982573465503732507429281522617283841154495314361036827133748055177} a^{3} + \frac{200118028504114333137493282680409858727805793355411405312193855927666380060}{1892766608947720396511197522287844567851851523463485943083110481401244165531} a^{2} - \frac{802498473492710478699293739091990606566612789690567483752489669621147303305}{1892766608947720396511197522287844567851851523463485943083110481401244165531} a + \frac{6049551139393777999813974780241142583975879028423499592098618480453099956}{99619295207774757711115659067781293044834290708604523320163709547433903449}$

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

Class group and class number

$C_{11}$, which has order $11$ (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:  $9$
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:  \( 2258908274.67 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T168:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 800
The 44 conjugacy class representatives for t20n168
Character table for t20n168 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.54025.1, 10.2.15138753125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{10}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
$71$71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.5.0.1$x^{5} - x + 8$$1$$5$$0$$C_5$$[\ ]^{5}$
71.5.0.1$x^{5} - x + 8$$1$$5$$0$$C_5$$[\ ]^{5}$
2161Data not computed