↑

Properties

Label 20.16.1818109779...0000.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{24}\cdot 5^{10}\cdot 13^{4}\cdot 29^{10}\cdot 31^{4}$
Root discriminant $91.83$
Ramified primes $2, 5, 13, 29, 31$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1028

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1862501, 16815772, 9187632, -67041885, -25985931, 94957418, 2051604, -57718272, 17250050, 10947276, -6177196, -405306, 785766, -66298, -39076, 4236, 525, 128, -12, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 - 12*x^18 + 128*x^17 + 525*x^16 + 4236*x^15 - 39076*x^14 - 66298*x^13 + 785766*x^12 - 405306*x^11 - 6177196*x^10 + 10947276*x^9 + 17250050*x^8 - 57718272*x^7 + 2051604*x^6 + 94957418*x^5 - 25985931*x^4 - 67041885*x^3 + 9187632*x^2 + 16815772*x + 1862501)
 
gp: K = bnfinit(x^20 - 9*x^19 - 12*x^18 + 128*x^17 + 525*x^16 + 4236*x^15 - 39076*x^14 - 66298*x^13 + 785766*x^12 - 405306*x^11 - 6177196*x^10 + 10947276*x^9 + 17250050*x^8 - 57718272*x^7 + 2051604*x^6 + 94957418*x^5 - 25985931*x^4 - 67041885*x^3 + 9187632*x^2 + 16815772*x + 1862501, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} - 12 x^{18} + 128 x^{17} + 525 x^{16} + 4236 x^{15} - 39076 x^{14} - 66298 x^{13} + 785766 x^{12} - 405306 x^{11} - 6177196 x^{10} + 10947276 x^{9} + 17250050 x^{8} - 57718272 x^{7} + 2051604 x^{6} + 94957418 x^{5} - 25985931 x^{4} - 67041885 x^{3} + 9187632 x^{2} + 16815772 x + 1862501 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1818109779441283591477572567040000000000=2^{24}\cdot 5^{10}\cdot 13^{4}\cdot 29^{10}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.83$
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:  $2, 5, 13, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{433845403382207196114153008165992692539633635122770911518259619579486733694} a^{19} - \frac{98570733935935596830083181593110166182710480366186827809749282076827448179}{433845403382207196114153008165992692539633635122770911518259619579486733694} a^{18} + \frac{3116811488036537365290216082602104221821979761301200097279302920316608483}{216922701691103598057076504082996346269816817561385455759129809789743366847} a^{17} + \frac{7035037741995278982686384586503073377983689354159436457036282853683263736}{216922701691103598057076504082996346269816817561385455759129809789743366847} a^{16} + \frac{106352599907685309221947737160147529169661823384007693179205513942089774629}{433845403382207196114153008165992692539633635122770911518259619579486733694} a^{15} - \frac{6440287521008297162650152893487389730275344640508589498921188567740911045}{216922701691103598057076504082996346269816817561385455759129809789743366847} a^{14} - \frac{23175767805246291320028880845928623558295408939103749716568453363311570112}{216922701691103598057076504082996346269816817561385455759129809789743366847} a^{13} - \frac{6000203876524895669910329155325262482832357430804988596949252154255825585}{216922701691103598057076504082996346269816817561385455759129809789743366847} a^{12} - \frac{164174036131245117420099986013605009894945236598348946400521982822751104053}{433845403382207196114153008165992692539633635122770911518259619579486733694} a^{11} + \frac{8507444865420473185041152276519338472546441714783896196126542373333527707}{25520317846012188006714882833293687796449037360162994795191742328205101982} a^{10} + \frac{8718445045895992148220055997546285823225468963335610001667727574309731565}{216922701691103598057076504082996346269816817561385455759129809789743366847} a^{9} - \frac{22193357088557647877849225584230036604462702533326119525582057718243651548}{216922701691103598057076504082996346269816817561385455759129809789743366847} a^{8} + \frac{15268613546648370231260811895008315543532527796866687083578685656322081549}{39440491216564290555832091651453881139966694102070082865296329052680612154} a^{7} - \frac{22027923878402860615356474875714788169612401606240113046390478160834080579}{216922701691103598057076504082996346269816817561385455759129809789743366847} a^{6} + \frac{5303926785366416580687641809016333597428802889901319227221963401129282425}{12760158923006094003357441416646843898224518680081497397595871164102550991} a^{5} - \frac{61936755197015679963641921510633262233911196731288443728873571887563708941}{216922701691103598057076504082996346269816817561385455759129809789743366847} a^{4} + \frac{787355471352857444619802036749118776661501780613096295199179225761306044}{216922701691103598057076504082996346269816817561385455759129809789743366847} a^{3} + \frac{2010083196874166455376013607867520773229639846255882681171462889508104877}{19720245608282145277916045825726940569983347051035041432648164526340306077} a^{2} - \frac{2167080275838320824134897193222339626298214092573712428041988801614237794}{216922701691103598057076504082996346269816817561385455759129809789743366847} a - \frac{34185448117006728657859167415812900552581320499923785504035698508438935222}{216922701691103598057076504082996346269816817561385455759129809789743366847}$

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:  $17$
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:  \( 5714348245800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.109268775200000.1

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

Sibling fields

Degree 20 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
$13$13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.6.5.2$x^{6} + 58$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
29.10.5.1$x^{10} - 1682 x^{6} + 707281 x^{2} - 2481849029$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.8.0.1$x^{8} - x + 22$$1$$8$$0$$C_8$$[\ ]^{8}$