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Properties

Label 20.2.18144464404...0000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{33}\cdot 3^{16}\cdot 5^{15}\cdot 7^{18}\cdot 11^{5}\cdot 19^{10}$
Root discriminant $1155.95$
Ramified primes $2, 3, 5, 7, 11, 19$
Class number Not computed
Class group Not computed
Galois group $D_4\times F_5$ (as 20T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3143510018924000, 0, 36520551394800, 0, 24804966751136, 0, 88440811800, 0, -20127194318, 0, 1515223395, 0, 11146009, 0, -783210, 0, 68, 0, 15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 15*x^18 + 68*x^16 - 783210*x^14 + 11146009*x^12 + 1515223395*x^10 - 20127194318*x^8 + 88440811800*x^6 + 24804966751136*x^4 + 36520551394800*x^2 - 3143510018924000)
 
gp: K = bnfinit(x^20 + 15*x^18 + 68*x^16 - 783210*x^14 + 11146009*x^12 + 1515223395*x^10 - 20127194318*x^8 + 88440811800*x^6 + 24804966751136*x^4 + 36520551394800*x^2 - 3143510018924000, 1)
 

Normalized defining polynomial

\( x^{20} + 15 x^{18} + 68 x^{16} - 783210 x^{14} + 11146009 x^{12} + 1515223395 x^{10} - 20127194318 x^{8} + 88440811800 x^{6} + 24804966751136 x^{4} + 36520551394800 x^{2} - 3143510018924000 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-18144464404222713605652682728648958381753368576000000000000000=-\,2^{33}\cdot 3^{16}\cdot 5^{15}\cdot 7^{18}\cdot 11^{5}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1155.95$
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:  $2, 3, 5, 7, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{28} a^{6} - \frac{3}{14} a^{4} - \frac{9}{28} a^{2} + \frac{3}{14}$, $\frac{1}{28} a^{7} - \frac{3}{14} a^{5} - \frac{9}{28} a^{3} + \frac{3}{14} a$, $\frac{1}{28} a^{8} - \frac{3}{28} a^{4} - \frac{3}{14} a^{2} + \frac{2}{7}$, $\frac{1}{28} a^{9} - \frac{3}{28} a^{5} - \frac{3}{14} a^{3} + \frac{2}{7} a$, $\frac{1}{4312} a^{10} + \frac{39}{4312} a^{8} + \frac{47}{4312} a^{6} - \frac{367}{4312} a^{4} + \frac{335}{1078} a^{2} + \frac{47}{98}$, $\frac{1}{8624} a^{11} - \frac{1}{8624} a^{10} + \frac{39}{8624} a^{9} - \frac{39}{8624} a^{8} + \frac{47}{8624} a^{7} - \frac{47}{8624} a^{6} - \frac{367}{8624} a^{5} + \frac{367}{8624} a^{4} - \frac{743}{2156} a^{3} + \frac{743}{2156} a^{2} - \frac{51}{196} a + \frac{51}{196}$, $\frac{1}{94864} a^{12} + \frac{3}{47432} a^{10} + \frac{75}{23716} a^{8} + \frac{61}{6776} a^{6} + \frac{18071}{94864} a^{4} - \frac{853}{2156} a^{2} - \frac{1}{196}$, $\frac{1}{94864} a^{13} - \frac{5}{94864} a^{11} - \frac{1}{8624} a^{10} - \frac{129}{94864} a^{9} - \frac{39}{8624} a^{8} + \frac{337}{94864} a^{7} - \frac{47}{8624} a^{6} + \frac{5527}{23716} a^{5} + \frac{367}{8624} a^{4} - \frac{5}{98} a^{3} + \frac{743}{2156} a^{2} + \frac{25}{98} a + \frac{51}{196}$, $\frac{1}{94864} a^{14} - \frac{1}{88} a^{8} + \frac{1}{176} a^{6} + \frac{1093}{6776} a^{4} - \frac{1}{308} a^{2} + \frac{13}{98}$, $\frac{1}{189728} a^{15} - \frac{1}{189728} a^{14} + \frac{15}{1232} a^{9} - \frac{15}{1232} a^{8} + \frac{1}{352} a^{7} - \frac{1}{352} a^{6} + \frac{367}{13552} a^{5} - \frac{367}{13552} a^{4} + \frac{241}{616} a^{3} - \frac{241}{616} a^{2} - \frac{57}{196} a + \frac{57}{196}$, $\frac{1}{29218112} a^{16} + \frac{103}{29218112} a^{14} + \frac{1}{260876} a^{12} - \frac{23}{298144} a^{10} + \frac{21311}{4174016} a^{8} + \frac{16213}{4174016} a^{6} - \frac{25549}{189728} a^{4} - \frac{27575}{60368} a^{2} - \frac{311}{2744}$, $\frac{1}{2746502528} a^{17} - \frac{2053}{2746502528} a^{15} - \frac{293}{98089376} a^{13} + \frac{631}{196178752} a^{11} - \frac{1}{8624} a^{10} - \frac{1154017}{392357504} a^{9} - \frac{39}{8624} a^{8} + \frac{36313}{8007296} a^{7} - \frac{47}{8624} a^{6} - \frac{3556367}{17834432} a^{5} + \frac{367}{8624} a^{4} - \frac{42741}{120736} a^{3} + \frac{743}{2156} a^{2} + \frac{14263}{257936} a + \frac{51}{196}$, $\frac{1}{21030871291624709158529056763266413215671040} a^{18} - \frac{137719449355894674846712140697763}{85840290986223302687873701074556788635392} a^{16} - \frac{8932671355714925678987232362207081581}{10515435645812354579264528381633206607835520} a^{14} - \frac{120918144226156505834618057437004259}{42920145493111651343936850537278394317696} a^{12} - \frac{23450577862234956338814005410409529253}{3004410184517815594075579537609487602238720} a^{10} - \frac{1369510934019799122097160692143557895633}{600882036903563118815115907521897520447744} a^{8} - \frac{49532471653223796216447493547841809543}{9754578521161738941803829667563271435840} a^{6} + \frac{4959457128338832878859146630699243379}{46225758949412495952455285658665406224} a^{4} + \frac{176399436600713907615652878766535528}{8817413435967480964956974131320105895} a^{2} - \frac{1719411010262711658058337326239477}{8128320546757956031731191429341552}$, $\frac{1}{9884509507063613304508656678735214211365388800} a^{19} - \frac{137719449355894674846712140697763}{40344936763524952263300639505041690658634240} a^{17} + \frac{11408358097731320426659586128535712894459}{4942254753531806652254328339367607105682694400} a^{15} - \frac{1}{189728} a^{14} - \frac{460071714531807818450438203608771483773}{141207278672337332921552238267645917305219840} a^{13} - \frac{38725057567666277121547514829314038407813}{1412072786723373329215522382676459173052198400} a^{11} - \frac{400775826750601178366898606957280678208839}{40344936763524952263300639505041690658634240} a^{9} - \frac{15}{1232} a^{8} - \frac{19968891193646811356020607703317946283421}{32092563334622121118534599606283163023913600} a^{7} - \frac{1}{352} a^{6} + \frac{92440249152112119131098574160931762029}{2172610670622387309765398425957274092528} a^{5} - \frac{367}{13552} a^{4} + \frac{2415057354284682286382048773146124525647}{16576737259618864214119111366881799082600} a^{3} - \frac{241}{616} a^{2} - \frac{258451236380175842077982020617982341}{764062131395247866982731994358105888} a + \frac{57}{196}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times F_5$ (as 20T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 25 conjugacy class representatives for $D_4\times F_5$
Character table for $D_4\times F_5$ is not computed

Intermediate fields

\(\Q(\sqrt{665}) \), 4.2.778316000.1, 5.1.388962000.4, 10.2.13111440193727783460000000.1

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 R R R R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.10.19.33$x^{10} - 6 x^{4} + 4 x^{2} - 14$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$3$3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
5Data not computed
$7$7.10.9.1$x^{10} - 7$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
7.10.9.1$x^{10} - 7$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$