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Properties

Label 20.0.94964489254...0000.8
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $177.37$
Ramified primes $2, 3, 5, 7, 11$
Class number $23286560$ (GRH)
Class group $[2, 4, 2910820]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![24916306801, 0, -1180449005, 0, 71316345, 0, 11330859, 0, 6823873, 0, 1822723, 0, 303689, 0, 31131, 0, 1937, 0, 67, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 67*x^18 + 1937*x^16 + 31131*x^14 + 303689*x^12 + 1822723*x^10 + 6823873*x^8 + 11330859*x^6 + 71316345*x^4 - 1180449005*x^2 + 24916306801)
 
gp: K = bnfinit(x^20 + 67*x^18 + 1937*x^16 + 31131*x^14 + 303689*x^12 + 1822723*x^10 + 6823873*x^8 + 11330859*x^6 + 71316345*x^4 - 1180449005*x^2 + 24916306801, 1)
 

Normalized defining polynomial

\( x^{20} + 67 x^{18} + 1937 x^{16} + 31131 x^{14} + 303689 x^{12} + 1822723 x^{10} + 6823873 x^{8} + 11330859 x^{6} + 71316345 x^{4} - 1180449005 x^{2} + 24916306801 \)

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:  \(949644892545940254829738055123773440000000000=2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $177.37$
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:  $2, 3, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4620=2^{2}\cdot 3\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{4620}(799,·)$, $\chi_{4620}(1,·)$, $\chi_{4620}(2309,·)$, $\chi_{4620}(841,·)$, $\chi_{4620}(2059,·)$, $\chi_{4620}(1681,·)$, $\chi_{4620}(2899,·)$, $\chi_{4620}(3989,·)$, $\chi_{4620}(3739,·)$, $\chi_{4620}(3319,·)$, $\chi_{4620}(1889,·)$, $\chi_{4620}(3611,·)$, $\chi_{4620}(421,·)$, $\chi_{4620}(1511,·)$, $\chi_{4620}(1469,·)$, $\chi_{4620}(629,·)$, $\chi_{4620}(3191,·)$, $\chi_{4620}(251,·)$, $\chi_{4620}(2941,·)$, $\chi_{4620}(4031,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{157849} a^{11} + \frac{44}{157849} a^{9} + \frac{704}{157849} a^{7} + \frac{4928}{157849} a^{5} + \frac{14080}{157849} a^{3} + \frac{11264}{157849} a$, $\frac{1}{51062415161} a^{12} + \frac{23253683328}{51062415161} a^{10} + \frac{11023859166}{51062415161} a^{8} + \frac{1146777913}{51062415161} a^{6} + \frac{13847633303}{51062415161} a^{4} - \frac{23367165451}{51062415161} a^{2} - \frac{112069}{323489}$, $\frac{1}{51062415161} a^{13} + \frac{52}{51062415161} a^{11} + \frac{9110098242}{51062415161} a^{9} + \frac{21589018290}{51062415161} a^{7} + \frac{3756070459}{51062415161} a^{5} - \frac{23021679199}{51062415161} a^{3} + \frac{3011375485}{51062415161} a$, $\frac{1}{51062415161} a^{14} + \frac{25416529050}{51062415161} a^{10} + \frac{10034908429}{51062415161} a^{8} - \frac{4813965856}{51062415161} a^{6} + \frac{22837616460}{51062415161} a^{4} - \frac{7393984927}{51062415161} a^{2} + \frac{4786}{323489}$, $\frac{1}{51062415161} a^{15} - \frac{1680}{51062415161} a^{11} + \frac{15080689851}{51062415161} a^{9} + \frac{1080700945}{2220105007} a^{7} - \frac{24783846208}{51062415161} a^{5} + \frac{24321200122}{51062415161} a^{3} + \frac{15915130321}{51062415161} a$, $\frac{1}{51062415161} a^{16} + \frac{18521082726}{51062415161} a^{10} + \frac{9282817172}{51062415161} a^{8} + \frac{12493686675}{51062415161} a^{6} + \frac{3883835746}{51062415161} a^{4} - \frac{24987983711}{51062415161} a^{2} - \frac{5322}{323489}$, $\frac{1}{51062415161} a^{17} + \frac{43520}{51062415161} a^{11} + \frac{11355734684}{51062415161} a^{9} - \frac{5402048294}{51062415161} a^{7} - \frac{19261478715}{51062415161} a^{5} - \frac{25465776964}{51062415161} a^{3} + \frac{19202659084}{51062415161} a$, $\frac{1}{51062415161} a^{18} + \frac{17063375983}{51062415161} a^{10} + \frac{18699900142}{51062415161} a^{8} + \frac{12005774983}{51062415161} a^{6} + \frac{15219021759}{51062415161} a^{4} - \frac{817259872}{51062415161} a^{2} - \frac{773}{323489}$, $\frac{1}{51062415161} a^{19} - \frac{21789}{51062415161} a^{11} - \frac{17215789572}{51062415161} a^{9} - \frac{958693670}{51062415161} a^{7} - \frac{24469843651}{51062415161} a^{5} - \frac{4794557127}{51062415161} a^{3} - \frac{3303855081}{51062415161} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{2910820}$, which has order $23286560$ (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:  \( 11184526.893275889 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-1155}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-21}, \sqrt{55})\), \(\Q(\zeta_{11})^+\), 10.0.896474439937004544.3, 10.0.30094051633627471875.1, 10.10.7545432611200000.1

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

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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$