Properties

Label 20.0.21991403617...3184.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 7^{6}\cdot 23^{2}\cdot 431^{4}$
Root discriminant $11.67$
Ramified primes $2, 7, 23, 431$
Class number $1$
Class group Trivial
Galois Group 20T994

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 3, 9, 20, 15, 1, 3, 14, -13, -22, 8, 22, -24, -2, 16, -2, -11, 11, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 11*x^18 - 11*x^17 - 2*x^16 + 16*x^15 - 2*x^14 - 24*x^13 + 22*x^12 + 8*x^11 - 22*x^10 - 13*x^9 + 14*x^8 + 3*x^7 + x^6 + 15*x^5 + 20*x^4 + 9*x^3 + 3*x^2 + 2*x + 1)
gp: K = bnfinit(x^20 - 5*x^19 + 11*x^18 - 11*x^17 - 2*x^16 + 16*x^15 - 2*x^14 - 24*x^13 + 22*x^12 + 8*x^11 - 22*x^10 - 13*x^9 + 14*x^8 + 3*x^7 + x^6 + 15*x^5 + 20*x^4 + 9*x^3 + 3*x^2 + 2*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 5 x^{19} \) \(\mathstrut +\mathstrut 11 x^{18} \) \(\mathstrut -\mathstrut 11 x^{17} \) \(\mathstrut -\mathstrut 2 x^{16} \) \(\mathstrut +\mathstrut 16 x^{15} \) \(\mathstrut -\mathstrut 2 x^{14} \) \(\mathstrut -\mathstrut 24 x^{13} \) \(\mathstrut +\mathstrut 22 x^{12} \) \(\mathstrut +\mathstrut 8 x^{11} \) \(\mathstrut -\mathstrut 22 x^{10} \) \(\mathstrut -\mathstrut 13 x^{9} \) \(\mathstrut +\mathstrut 14 x^{8} \) \(\mathstrut +\mathstrut 3 x^{7} \) \(\mathstrut +\mathstrut x^{6} \) \(\mathstrut +\mathstrut 15 x^{5} \) \(\mathstrut +\mathstrut 20 x^{4} \) \(\mathstrut +\mathstrut 9 x^{3} \) \(\mathstrut +\mathstrut 3 x^{2} \) \(\mathstrut +\mathstrut 2 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(2199140361717170013184=2^{10}\cdot 7^{6}\cdot 23^{2}\cdot 431^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.67$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 7, 23, 431$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{567947447479} a^{19} - \frac{188290224203}{567947447479} a^{18} - \frac{174225288284}{567947447479} a^{17} + \frac{12253238385}{567947447479} a^{16} + \frac{18966534861}{567947447479} a^{15} - \frac{138182773363}{567947447479} a^{14} - \frac{257287552850}{567947447479} a^{13} - \frac{2825698004}{567947447479} a^{12} + \frac{214722052102}{567947447479} a^{11} + \frac{50164800310}{567947447479} a^{10} - \frac{152349283644}{567947447479} a^{9} + \frac{176256545782}{567947447479} a^{8} - \frac{30458889122}{567947447479} a^{7} - \frac{67544896520}{567947447479} a^{6} - \frac{129936515290}{567947447479} a^{5} + \frac{41891468240}{567947447479} a^{4} + \frac{13141853506}{567947447479} a^{3} - \frac{245090998797}{567947447479} a^{2} - \frac{238123711600}{567947447479} a - \frac{249453842442}{567947447479}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( \frac{68936272252}{567947447479} a^{19} - \frac{587143183448}{567947447479} a^{18} + \frac{2117265143094}{567947447479} a^{17} - \frac{4183267805908}{567947447479} a^{16} + \frac{4254724686062}{567947447479} a^{15} - \frac{231585731999}{567947447479} a^{14} - \frac{4207437433960}{567947447479} a^{13} + \frac{1419984163961}{567947447479} a^{12} + \frac{6742805600484}{567947447479} a^{11} - \frac{8950229774136}{567947447479} a^{10} + \frac{1365047554024}{567947447479} a^{9} + \frac{4662797225093}{567947447479} a^{8} + \frac{19788497056}{567947447479} a^{7} - \frac{3563299676417}{567947447479} a^{6} + \frac{1726772640216}{567947447479} a^{5} - \frac{163034697879}{567947447479} a^{4} - \frac{1236450365418}{567947447479} a^{3} - \frac{1891679423009}{567947447479} a^{2} + \frac{309646581486}{567947447479} a + \frac{43368248979}{567947447479} \),  \( \frac{293504572591}{567947447479} a^{19} - \frac{1704812644334}{567947447479} a^{18} + \frac{4336290135033}{567947447479} a^{17} - \frac{5222424524662}{567947447479} a^{16} - \frac{150813449678}{567947447479} a^{15} + \frac{9617111611187}{567947447479} a^{14} - \frac{9701653029943}{567947447479} a^{13} - \frac{3889288738888}{567947447479} a^{12} + \frac{13336426188188}{567947447479} a^{11} - \frac{3530194849954}{567947447479} a^{10} - \frac{12507855940696}{567947447479} a^{9} + \frac{8178727076943}{567947447479} a^{8} + \frac{3620815707770}{567947447479} a^{7} - \frac{2374922844801}{567947447479} a^{6} - \frac{1590117886975}{567947447479} a^{5} + \frac{5919555989051}{567947447479} a^{4} + \frac{955832415281}{567947447479} a^{3} - \frac{1307518715679}{567947447479} a^{2} - \frac{394787649623}{567947447479} a + \frac{662156881518}{567947447479} \),  \( \frac{158365982752}{567947447479} a^{19} - \frac{1241622898876}{567947447479} a^{18} + \frac{4404719589920}{567947447479} a^{17} - \frac{9150120090384}{567947447479} a^{16} + \frac{11425280313166}{567947447479} a^{15} - \frac{6963727576851}{567947447479} a^{14} - \frac{312340286963}{567947447479} a^{13} + \frac{72714588246}{567947447479} a^{12} + \frac{7776098937862}{567947447479} a^{11} - \frac{12314697346966}{567947447479} a^{10} + \frac{7479283188305}{567947447479} a^{9} - \frac{2417325969156}{567947447479} a^{8} + \frac{4757696478615}{567947447479} a^{7} - \frac{4288327603196}{567947447479} a^{6} + \frac{3202302708368}{567947447479} a^{5} - \frac{1622823035296}{567947447479} a^{4} - \frac{1285804128322}{567947447479} a^{3} - \frac{2578126148794}{567947447479} a^{2} - \frac{508076004183}{567947447479} a - \frac{766903201165}{567947447479} \),  \( \frac{487660646241}{567947447479} a^{19} - \frac{2714458694554}{567947447479} a^{18} + \frac{6978391276066}{567947447479} a^{17} - \frac{9653562081508}{567947447479} a^{16} + \frac{5073129967862}{567947447479} a^{15} + \frac{4736952588582}{567947447479} a^{14} - \frac{4654172184820}{567947447479} a^{13} - \frac{7441782258858}{567947447479} a^{12} + \frac{15021861727732}{567947447479} a^{11} - \frac{6062759994278}{567947447479} a^{10} - \frac{7340362063133}{567947447479} a^{9} - \frac{160275627541}{567947447479} a^{8} + \frac{5521871973688}{567947447479} a^{7} - \frac{3626069384330}{567947447479} a^{6} + \frac{2430687681321}{567947447479} a^{5} + \frac{7966630250872}{567947447479} a^{4} + \frac{5302683238336}{567947447479} a^{3} + \frac{2638416515570}{567947447479} a^{2} + \frac{2051714839778}{567947447479} a + \frac{1070828353584}{567947447479} \),  \( a \),  \( \frac{105511272117}{567947447479} a^{19} - \frac{700060620917}{567947447479} a^{18} + \frac{2274242579151}{567947447479} a^{17} - \frac{4491201545370}{567947447479} a^{16} + \frac{5441443801948}{567947447479} a^{15} - \frac{3237609605119}{567947447479} a^{14} - \frac{243491085354}{567947447479} a^{13} + \frac{932030175769}{567947447479} a^{12} + \frac{2430514278548}{567947447479} a^{11} - \frac{5314227200785}{567947447479} a^{10} + \frac{3757355697435}{567947447479} a^{9} - \frac{1547783971598}{567947447479} a^{8} + \frac{682809465574}{567947447479} a^{7} - \frac{1289296337021}{567947447479} a^{6} + \frac{799137447853}{567947447479} a^{5} + \frac{1099272455264}{567947447479} a^{4} + \frac{988252572478}{567947447479} a^{3} + \frac{308046095328}{567947447479} a^{2} + \frac{381188507360}{567947447479} a + \frac{362654778810}{567947447479} \),  \( \frac{860131700202}{567947447479} a^{19} - \frac{4907115925957}{567947447479} a^{18} + \frac{12763738213433}{567947447479} a^{17} - \frac{17486434325192}{567947447479} a^{16} + \frac{7846858718881}{567947447479} a^{15} + \frac{12540269639065}{567947447479} a^{14} - \frac{13584944156328}{567947447479} a^{13} - \frac{12393337854399}{567947447479} a^{12} + \frac{30670497307325}{567947447479} a^{11} - \frac{13062329544587}{567947447479} a^{10} - \frac{16589309780021}{567947447479} a^{9} + \frac{5552387716280}{567947447479} a^{8} + \frac{9599621250822}{567947447479} a^{7} - \frac{5765487235215}{567947447479} a^{6} + \frac{3197111621079}{567947447479} a^{5} + \frac{12421469077706}{567947447479} a^{4} + \frac{7416722368587}{567947447479} a^{3} + \frac{894351039685}{567947447479} a^{2} + \frac{1173428128661}{567947447479} a + \frac{1250239218663}{567947447479} \),  \( \frac{28088777574}{567947447479} a^{19} - \frac{24382680215}{567947447479} a^{18} - \frac{427679203109}{567947447479} a^{17} + \frac{1825023181737}{567947447479} a^{16} - \frac{3509679186242}{567947447479} a^{15} + \frac{3199964570629}{567947447479} a^{14} + \frac{269403529900}{567947447479} a^{13} - \frac{2516812877846}{567947447479} a^{12} - \frac{281460870403}{567947447479} a^{11} + \frac{4138457426655}{567947447479} a^{10} - \frac{3536555395580}{567947447479} a^{9} - \frac{619734376528}{567947447479} a^{8} + \frac{19741141103}{567947447479} a^{7} + \frac{1961339118396}{567947447479} a^{6} - \frac{50384003732}{567947447479} a^{5} + \frac{478416946721}{567947447479} a^{4} + \frac{1346041852158}{567947447479} a^{3} + \frac{326474834375}{567947447479} a^{2} - \frac{1064497393267}{567947447479} a - \frac{182976152001}{567947447479} \),  \( \frac{269456357483}{567947447479} a^{19} - \frac{2063267299240}{567947447479} a^{18} + \frac{7127506917069}{567947447479} a^{17} - \frac{14089457937422}{567947447479} a^{16} + \frac{15569482286613}{567947447479} a^{15} - \frac{5091856461196}{567947447479} a^{14} - \frac{7939312548219}{567947447479} a^{13} + \frac{3928029004312}{567947447479} a^{12} + \frac{14567038421767}{567947447479} a^{11} - \frac{21781042462054}{567947447479} a^{10} + \frac{6887744750984}{567947447479} a^{9} + \frac{5342023633679}{567947447479} a^{8} + \frac{2748555486474}{567947447479} a^{7} - \frac{7867297966506}{567947447479} a^{6} + \frac{4111815515067}{567947447479} a^{5} + \frac{1315291911838}{567947447479} a^{4} - \frac{3816330034824}{567947447479} a^{3} - \frac{3085772263482}{567947447479} a^{2} + \frac{535158346358}{567947447479} a - \frac{396857135885}{567947447479} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 166.624941197 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

20T994:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 1966080
The 280 conjugacy class representatives for t20n994 are not computed
Character table for t20n994 is not computed

Intermediate fields

5.1.3017.1, 10.0.209352647.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.3$x^{10} - 9 x^{8} + 22 x^{6} - 46 x^{4} + 9 x^{2} - 9$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
$7$7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$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$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
431Data not computed