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Properties

Label 20.0.12719384508...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{10}\cdot 5^{22}\cdot 13^{10}$
Root discriminant $63.86$
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![26214400, 0, -8192000, 0, 204800, 0, 448000, 0, 134800, 0, -79625, 0, 17685, 0, -2850, 0, 330, 0, -25, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 25*x^18 + 330*x^16 - 2850*x^14 + 17685*x^12 - 79625*x^10 + 134800*x^8 + 448000*x^6 + 204800*x^4 - 8192000*x^2 + 26214400)
 
gp: K = bnfinit(x^20 - 25*x^18 + 330*x^16 - 2850*x^14 + 17685*x^12 - 79625*x^10 + 134800*x^8 + 448000*x^6 + 204800*x^4 - 8192000*x^2 + 26214400, 1)
 

Normalized defining polynomial

\( x^{20} - 25 x^{18} + 330 x^{16} - 2850 x^{14} + 17685 x^{12} - 79625 x^{10} + 134800 x^{8} + 448000 x^{6} + 204800 x^{4} - 8192000 x^{2} + 26214400 \)

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:  \(1271938450811187656250000000000000000=2^{16}\cdot 3^{10}\cdot 5^{22}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.86$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{20} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{20} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{80} a^{12} - \frac{1}{80} a^{10} - \frac{1}{8} a^{8} + \frac{1}{8} a^{6} - \frac{3}{16} a^{4} + \frac{7}{16} a^{2} - \frac{1}{2} a$, $\frac{1}{320} a^{13} + \frac{7}{320} a^{11} + \frac{1}{32} a^{9} + \frac{3}{32} a^{7} + \frac{17}{64} a^{5} - \frac{21}{64} a^{3} + \frac{1}{4} a$, $\frac{1}{1280} a^{14} + \frac{7}{1280} a^{12} - \frac{11}{640} a^{10} + \frac{3}{128} a^{8} + \frac{17}{256} a^{6} - \frac{1}{2} a^{5} + \frac{107}{256} a^{4} - \frac{1}{2} a^{3} - \frac{1}{16} a^{2}$, $\frac{1}{10240} a^{15} - \frac{1}{2560} a^{14} - \frac{9}{10240} a^{13} + \frac{9}{2560} a^{12} - \frac{7}{1024} a^{11} - \frac{29}{1280} a^{10} - \frac{13}{1024} a^{9} + \frac{13}{256} a^{8} + \frac{49}{2048} a^{7} - \frac{49}{512} a^{6} - \frac{805}{2048} a^{5} + \frac{37}{512} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{81920} a^{16} - \frac{5}{16384} a^{14} - \frac{1}{640} a^{13} + \frac{37}{40960} a^{12} + \frac{9}{640} a^{11} - \frac{273}{40960} a^{10} + \frac{7}{64} a^{9} - \frac{1071}{16384} a^{8} - \frac{3}{64} a^{7} + \frac{3019}{16384} a^{6} - \frac{17}{128} a^{5} - \frac{379}{1024} a^{4} - \frac{59}{128} a^{3} - \frac{13}{64} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{327680} a^{17} + \frac{7}{327680} a^{15} - \frac{107}{163840} a^{13} - \frac{3441}{163840} a^{11} - \frac{5999}{65536} a^{9} + \frac{491}{65536} a^{7} - \frac{1221}{4096} a^{5} - \frac{77}{256} a^{3} - \frac{1}{2} a^{2} - \frac{7}{16} a - \frac{1}{2}$, $\frac{1}{1305824125141279047680} a^{18} + \frac{190307051709063}{1305824125141279047680} a^{16} - \frac{3481468792380399}{130582412514127904768} a^{14} - \frac{2444111659759686897}{652912062570639523840} a^{12} - \frac{1}{40} a^{11} + \frac{600230957244966801}{261164825028255809536} a^{10} - \frac{1}{8} a^{9} + \frac{27475337559435861099}{261164825028255809536} a^{8} + \frac{1112853248528954739}{16322801564265988096} a^{6} - \frac{351339661756303469}{1020175097766624256} a^{4} + \frac{1}{8} a^{3} - \frac{6156042104174163}{63760943610414016} a^{2} - \frac{3}{8} a + \frac{305719343202824}{996264743912719}$, $\frac{1}{5223296500565116190720} a^{19} + \frac{190307051709063}{5223296500565116190720} a^{17} - \frac{3481468792380399}{522329650056511619072} a^{15} - \frac{2444111659759686897}{2611648250282558095360} a^{13} - \frac{1}{160} a^{12} + \frac{68292361043288786389}{5223296500565116190720} a^{11} + \frac{1}{160} a^{10} - \frac{103107074954692043669}{1044659300113023238144} a^{9} - \frac{1}{16} a^{8} + \frac{1112853248528954739}{65291206257063952384} a^{7} - \frac{3}{16} a^{6} + \frac{1689010533776945043}{4080700391066497024} a^{5} + \frac{15}{32} a^{4} - \frac{85857221617191683}{255043774441656064} a^{3} + \frac{13}{32} a^{2} + \frac{76429835800706}{996264743912719} a$

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:  \( 12384758684.151869 \) (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

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-15}, \sqrt{-39})\), 5.1.50000.1, 10.0.3037500000000.2, 10.0.225560497500000000.1, 10.2.4641162500000000.2

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}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.10.11.7$x^{10} + 5 x^{2} + 10$$10$$1$$11$$F_{5}\times C_2$$[5/4]_{4}^{2}$
5.10.11.7$x^{10} + 5 x^{2} + 10$$10$$1$$11$$F_{5}\times C_2$$[5/4]_{4}^{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$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}$