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Properties

Label 20.0.73038164166...9296.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{16}\cdot 23^{6}$
Root discriminant $49.34$
Ramified primes $2, 11, 23$
Class number $160$ (GRH)
Class group $[2, 4, 20]$ (GRH)
Galois group $C_2\times C_2^4:C_5$ (as 20T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![612679, -3882390, 10953478, -18640836, 22096203, -19938342, 14259924, -8011944, 3400561, -1013022, 156658, 26478, -22332, 4518, 806, -870, 351, -90, 24, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 24*x^18 - 90*x^17 + 351*x^16 - 870*x^15 + 806*x^14 + 4518*x^13 - 22332*x^12 + 26478*x^11 + 156658*x^10 - 1013022*x^9 + 3400561*x^8 - 8011944*x^7 + 14259924*x^6 - 19938342*x^5 + 22096203*x^4 - 18640836*x^3 + 10953478*x^2 - 3882390*x + 612679)
 
gp: K = bnfinit(x^20 - 6*x^19 + 24*x^18 - 90*x^17 + 351*x^16 - 870*x^15 + 806*x^14 + 4518*x^13 - 22332*x^12 + 26478*x^11 + 156658*x^10 - 1013022*x^9 + 3400561*x^8 - 8011944*x^7 + 14259924*x^6 - 19938342*x^5 + 22096203*x^4 - 18640836*x^3 + 10953478*x^2 - 3882390*x + 612679, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 24 x^{18} - 90 x^{17} + 351 x^{16} - 870 x^{15} + 806 x^{14} + 4518 x^{13} - 22332 x^{12} + 26478 x^{11} + 156658 x^{10} - 1013022 x^{9} + 3400561 x^{8} - 8011944 x^{7} + 14259924 x^{6} - 19938342 x^{5} + 22096203 x^{4} - 18640836 x^{3} + 10953478 x^{2} - 3882390 x + 612679 \)

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:  \(7303816416639774784171798530359296=2^{30}\cdot 11^{16}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.34$
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 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}{30337233704259289141020555282997062727101040965490643910791} a^{19} + \frac{9940272187269595702187103983979092943773450255040432311656}{30337233704259289141020555282997062727101040965490643910791} a^{18} - \frac{6933030717091906894833740854849771335829978650117559415949}{30337233704259289141020555282997062727101040965490643910791} a^{17} - \frac{11657546393775166467855204409139828979144205283605464683174}{30337233704259289141020555282997062727101040965490643910791} a^{16} - \frac{34876169383092551199896730979696872226925945155528396828}{30337233704259289141020555282997062727101040965490643910791} a^{15} + \frac{13989214903911046005181304204501965925043862079925087368046}{30337233704259289141020555282997062727101040965490643910791} a^{14} + \frac{5026783863201423961016117322025131775491840016994202378037}{30337233704259289141020555282997062727101040965490643910791} a^{13} + \frac{12852336354542203549544399331917014947198161165145039195383}{30337233704259289141020555282997062727101040965490643910791} a^{12} + \frac{13411909290990231571037331776081625322866613524189942332370}{30337233704259289141020555282997062727101040965490643910791} a^{11} - \frac{25370689905441986325952580531222098576785606241104390720}{72403899055511429930836647453453610327210121636015856589} a^{10} - \frac{11636210609654990102740591268241351924301349276139256766059}{30337233704259289141020555282997062727101040965490643910791} a^{9} + \frac{7741616612804290362021749449604906246755828169080774676692}{30337233704259289141020555282997062727101040965490643910791} a^{8} + \frac{12014819278608523220228065051126676255643779768505169736667}{30337233704259289141020555282997062727101040965490643910791} a^{7} - \frac{10028678310747896346922836981498794763288429548598805626583}{30337233704259289141020555282997062727101040965490643910791} a^{6} - \frac{7184762842503385252768195508448693609035032103134762799537}{30337233704259289141020555282997062727101040965490643910791} a^{5} - \frac{1673550754756718564761706363297096152977537620965688009562}{30337233704259289141020555282997062727101040965490643910791} a^{4} - \frac{2106371744415528067432945650066085191216335572115681107971}{30337233704259289141020555282997062727101040965490643910791} a^{3} + \frac{12719598732893983049667935374169237984229234518026627057891}{30337233704259289141020555282997062727101040965490643910791} a^{2} + \frac{12441277149180393100556557825056189664808912295160678980020}{30337233704259289141020555282997062727101040965490643910791} a + \frac{4329613442550166499448011345691442888896158523956193707818}{30337233704259289141020555282997062727101040965490643910791}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{20}$, which has order $160$ (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:  \( 2545371.69018 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 16 conjugacy class representatives for $C_2\times C_2^4:C_5$
Character table for $C_2\times C_2^4:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.0.2670699013250048.1, 10.0.5048580365312.1, 10.10.116117348402176.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{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}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$