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Properties

Label 20.0.21363681729...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 13^{10}$
Root discriminant $52.06$
Ramified primes $2, 3, 5, 13$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4194304, 0, -5242880, 3538944, 3391488, -2820096, -741888, 861696, 127104, -108792, -44106, 4704, 11928, -2748, -66, 27, -75, 60, -8, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 8*x^18 + 60*x^17 - 75*x^16 + 27*x^15 - 66*x^14 - 2748*x^13 + 11928*x^12 + 4704*x^11 - 44106*x^10 - 108792*x^9 + 127104*x^8 + 861696*x^7 - 741888*x^6 - 2820096*x^5 + 3391488*x^4 + 3538944*x^3 - 5242880*x^2 + 4194304)
 
gp: K = bnfinit(x^20 - 3*x^19 - 8*x^18 + 60*x^17 - 75*x^16 + 27*x^15 - 66*x^14 - 2748*x^13 + 11928*x^12 + 4704*x^11 - 44106*x^10 - 108792*x^9 + 127104*x^8 + 861696*x^7 - 741888*x^6 - 2820096*x^5 + 3391488*x^4 + 3538944*x^3 - 5242880*x^2 + 4194304, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 8 x^{18} + 60 x^{17} - 75 x^{16} + 27 x^{15} - 66 x^{14} - 2748 x^{13} + 11928 x^{12} + 4704 x^{11} - 44106 x^{10} - 108792 x^{9} + 127104 x^{8} + 861696 x^{7} - 741888 x^{6} - 2820096 x^{5} + 3391488 x^{4} + 3538944 x^{3} - 5242880 x^{2} + 4194304 \)

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:  \(21363681729976837664400000000000000=2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.06$
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:  $2, 3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{3}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a$, $\frac{1}{32} a^{12} + \frac{1}{32} a^{11} - \frac{1}{8} a^{10} + \frac{3}{8} a^{9} + \frac{5}{32} a^{8} + \frac{15}{32} a^{7} - \frac{3}{16} a^{6} + \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{5}{16} a^{2}$, $\frac{1}{128} a^{13} + \frac{1}{128} a^{12} - \frac{1}{32} a^{11} - \frac{5}{32} a^{10} + \frac{37}{128} a^{9} + \frac{47}{128} a^{8} - \frac{3}{64} a^{7} - \frac{5}{32} a^{6} + \frac{1}{16} a^{5} + \frac{27}{64} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{512} a^{14} + \frac{1}{512} a^{13} - \frac{1}{128} a^{12} - \frac{5}{128} a^{11} + \frac{37}{512} a^{10} + \frac{175}{512} a^{9} - \frac{67}{256} a^{8} - \frac{5}{128} a^{7} - \frac{15}{64} a^{6} - \frac{1}{2} a^{5} + \frac{91}{256} a^{4} - \frac{1}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{2048} a^{15} + \frac{1}{2048} a^{14} - \frac{1}{512} a^{13} - \frac{5}{512} a^{12} + \frac{37}{2048} a^{11} - \frac{337}{2048} a^{10} - \frac{67}{1024} a^{9} - \frac{133}{512} a^{8} - \frac{15}{256} a^{7} - \frac{3}{8} a^{6} + \frac{347}{1024} a^{5} + \frac{31}{64} a^{4} + \frac{1}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{8192} a^{16} + \frac{1}{8192} a^{15} - \frac{1}{2048} a^{14} - \frac{5}{2048} a^{13} + \frac{37}{8192} a^{12} - \frac{337}{8192} a^{11} + \frac{957}{4096} a^{10} - \frac{645}{2048} a^{9} + \frac{497}{1024} a^{8} + \frac{13}{32} a^{7} - \frac{677}{4096} a^{6} - \frac{97}{256} a^{5} - \frac{1}{4} a^{4} - \frac{15}{64} a^{3} + \frac{7}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{32768} a^{17} + \frac{1}{32768} a^{16} - \frac{1}{8192} a^{15} - \frac{5}{8192} a^{14} + \frac{37}{32768} a^{13} - \frac{337}{32768} a^{12} + \frac{957}{16384} a^{11} + \frac{1403}{8192} a^{10} + \frac{497}{4096} a^{9} - \frac{51}{128} a^{8} + \frac{3419}{16384} a^{7} - \frac{353}{1024} a^{6} - \frac{1}{16} a^{5} - \frac{15}{256} a^{4} - \frac{25}{64} a^{3} + \frac{7}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{2597453824} a^{18} + \frac{21141}{2597453824} a^{17} - \frac{3907}{162340864} a^{16} + \frac{69527}{649363456} a^{15} + \frac{584469}{2597453824} a^{14} + \frac{9911059}{2597453824} a^{13} - \frac{2330571}{185532416} a^{12} - \frac{9460947}{649363456} a^{11} - \frac{63066337}{324681728} a^{10} + \frac{7801581}{81170432} a^{9} - \frac{4686183}{68354048} a^{8} + \frac{38961}{17088512} a^{7} - \frac{5016311}{10146304} a^{6} - \frac{6181013}{20292608} a^{5} - \frac{4507}{39634} a^{4} + \frac{73709}{634144} a^{3} + \frac{16367}{39634} a^{2} - \frac{23721}{79268} a - \frac{9396}{19817}$, $\frac{1}{5357657758539687749433901114767668936704} a^{19} + \frac{168369273543963927558808884525}{5357657758539687749433901114767668936704} a^{18} - \frac{3536833307409336077383462977400309}{669707219817460968679237639345958617088} a^{17} - \frac{65658590781554585001797445772452357}{1339414439634921937358475278691917234176} a^{16} - \frac{1062961161572156256808825930864603531}{5357657758539687749433901114767668936704} a^{15} + \frac{2384829666810764278105670738629299531}{5357657758539687749433901114767668936704} a^{14} - \frac{5441957392815171407702788300326628289}{2678828879269843874716950557383834468352} a^{13} + \frac{576136201460671300083185391369930381}{1339414439634921937358475278691917234176} a^{12} - \frac{149021040639478196530148465063909757}{5035392630206473448716072476285403136} a^{11} + \frac{3133527615005675580501556823927424981}{167426804954365242169809409836489654272} a^{10} + \frac{797100008025612268678343514785825071131}{2678828879269843874716950557383834468352} a^{9} - \frac{8857173984587561756966609998169972505}{35247748411445314141012507333997821952} a^{8} + \frac{42415745887166821927559909653550891}{168947330932760082916053894890504192} a^{7} + \frac{974710591153857714365796217871251713}{5979528748370187220350336065588916224} a^{6} + \frac{3631039692957516600191192992174539253}{10464175309647827635613088114780603392} a^{5} + \frac{650032557665953033751742118877249493}{1308021913705978454451636014347575424} a^{4} + \frac{58070486942048836235007099119445583}{654010956852989227225818007173787712} a^{3} + \frac{14831282214298307565169531089148369}{163502739213247306806454501793446928} a^{2} + \frac{226880262242471256417501007797172}{10218921200827956675403406362090433} a - \frac{2760498962973901432865552766307860}{10218921200827956675403406362090433}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( 1756850656.1159236 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{-39}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-15}, \sqrt{-39})\), 5.1.162000.1, 10.0.393660000000.1, 10.0.29232640476000000.1, 10.2.48721067460000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3Data not computed
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
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}$