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Properties

Label 20.4.26062314028...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 3^{10}\cdot 5^{22}\cdot 7^{10}$
Root discriminant $46.86$
Ramified primes $2, 3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![58105324, 5650100, -78750420, -1739160, 48188040, -205998, -18138180, 158310, 4772700, -29740, -924247, 3010, 132625, -130, -13780, -4, 990, 0, -45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 45*x^18 + 990*x^16 - 4*x^15 - 13780*x^14 - 130*x^13 + 132625*x^12 + 3010*x^11 - 924247*x^10 - 29740*x^9 + 4772700*x^8 + 158310*x^7 - 18138180*x^6 - 205998*x^5 + 48188040*x^4 - 1739160*x^3 - 78750420*x^2 + 5650100*x + 58105324)
 
gp: K = bnfinit(x^20 - 45*x^18 + 990*x^16 - 4*x^15 - 13780*x^14 - 130*x^13 + 132625*x^12 + 3010*x^11 - 924247*x^10 - 29740*x^9 + 4772700*x^8 + 158310*x^7 - 18138180*x^6 - 205998*x^5 + 48188040*x^4 - 1739160*x^3 - 78750420*x^2 + 5650100*x + 58105324, 1)
 

Normalized defining polynomial

\( x^{20} - 45 x^{18} + 990 x^{16} - 4 x^{15} - 13780 x^{14} - 130 x^{13} + 132625 x^{12} + 3010 x^{11} - 924247 x^{10} - 29740 x^{9} + 4772700 x^{8} + 158310 x^{7} - 18138180 x^{6} - 205998 x^{5} + 48188040 x^{4} - 1739160 x^{3} - 78750420 x^{2} + 5650100 x + 58105324 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2606231402843906250000000000000000=2^{16}\cdot 3^{10}\cdot 5^{22}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.86$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9}$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} + \frac{1}{7} a^{16} - \frac{1}{14} a^{15} - \frac{1}{7} a^{14} - \frac{1}{14} a^{13} + \frac{3}{14} a^{10} + \frac{5}{14} a^{9} + \frac{3}{7} a^{8} - \frac{1}{2} a^{7} + \frac{3}{7} a^{6} + \frac{1}{14} a^{5} + \frac{2}{7} a^{4} - \frac{2}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{121273819132579385744869648546667403107492188448490516732086} a^{19} + \frac{2244736947959549821226155390766804891552429453100645996767}{121273819132579385744869648546667403107492188448490516732086} a^{18} - \frac{10596446067343116505143722817220965089641278796984933112993}{121273819132579385744869648546667403107492188448490516732086} a^{17} - \frac{8772615621907028938928360689258946603782603690598721262835}{121273819132579385744869648546667403107492188448490516732086} a^{16} + \frac{175811739488554363677957637438269185030527181458018760839}{1237487950332442711682343352517014317423389678045821599307} a^{15} - \frac{7140002498155017668483658241312952696893260525788786985369}{60636909566289692872434824273333701553746094224245258366043} a^{14} + \frac{3481521752576381412927075901525490649660216592955918981493}{121273819132579385744869648546667403107492188448490516732086} a^{13} - \frac{730620319016944946695896190744750309665978663289977147547}{17324831304654197963552806935238200443927455492641502390298} a^{12} - \frac{6992697565892467031350371559347090073135943043355817336875}{60636909566289692872434824273333701553746094224245258366043} a^{11} - \frac{603839297282414274817146168764250659873530265325194891917}{60636909566289692872434824273333701553746094224245258366043} a^{10} + \frac{12879827338688686105640348620614741830008604655618677380144}{60636909566289692872434824273333701553746094224245258366043} a^{9} - \frac{29753876733648390692672910931557922847082978433256705696223}{60636909566289692872434824273333701553746094224245258366043} a^{8} - \frac{55743742887026756455640301435725438856338950516872081562837}{121273819132579385744869648546667403107492188448490516732086} a^{7} + \frac{50811207968060326504590722454286864732771400035394383035011}{121273819132579385744869648546667403107492188448490516732086} a^{6} + \frac{3096694902764487603643915704169178791901624059288800105096}{60636909566289692872434824273333701553746094224245258366043} a^{5} + \frac{6222559820613699490989247721732368881132442458773116234614}{60636909566289692872434824273333701553746094224245258366043} a^{4} - \frac{10534288228136499826111757528085325966242242394598825497074}{60636909566289692872434824273333701553746094224245258366043} a^{3} - \frac{20816862875273251705295072466801729321006437209565679907801}{60636909566289692872434824273333701553746094224245258366043} a^{2} - \frac{3003550740394755660949577455841962900954982454886923594947}{60636909566289692872434824273333701553746094224245258366043} a - \frac{22567424494664241836349375660074317915399134816967121600845}{60636909566289692872434824273333701553746094224245258366043}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 1303014860.790379 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}, \sqrt{21})\), 5.1.50000.1, 10.2.12500000000.1, 10.2.10210252500000000.1, 10.2.51051262500000000.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{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$$[\ ]_{2}^{4}$
$5$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$