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Properties

Label 20.4.65735505203...0000.2
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 5^{22}\cdot 29^{10}$
Root discriminant $55.07$
Ramified primes $2, 5, 29$
Class number Not computed
Class group Not computed
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![1055258980, 42769300, -1110770180, -10809480, 536314500, -704030, -159124680, 545030, 32453600, -83460, -4761951, 6210, 507405, -170, -38620, -4, 2010, 0, -65, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 65*x^18 + 2010*x^16 - 4*x^15 - 38620*x^14 - 170*x^13 + 507405*x^12 + 6210*x^11 - 4761951*x^10 - 83460*x^9 + 32453600*x^8 + 545030*x^7 - 159124680*x^6 - 704030*x^5 + 536314500*x^4 - 10809480*x^3 - 1110770180*x^2 + 42769300*x + 1055258980)
 
gp: K = bnfinit(x^20 - 65*x^18 + 2010*x^16 - 4*x^15 - 38620*x^14 - 170*x^13 + 507405*x^12 + 6210*x^11 - 4761951*x^10 - 83460*x^9 + 32453600*x^8 + 545030*x^7 - 159124680*x^6 - 704030*x^5 + 536314500*x^4 - 10809480*x^3 - 1110770180*x^2 + 42769300*x + 1055258980, 1)
 

Normalized defining polynomial

\( x^{20} - 65 x^{18} + 2010 x^{16} - 4 x^{15} - 38620 x^{14} - 170 x^{13} + 507405 x^{12} + 6210 x^{11} - 4761951 x^{10} - 83460 x^{9} + 32453600 x^{8} + 545030 x^{7} - 159124680 x^{6} - 704030 x^{5} + 536314500 x^{4} - 10809480 x^{3} - 1110770180 x^{2} + 42769300 x + 1055258980 \)

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:  \(65735505203156406250000000000000000=2^{16}\cdot 5^{22}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.07$
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:  $2, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9}$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} - \frac{1}{7} a^{16} - \frac{1}{7} a^{15} - \frac{3}{14} a^{14} + \frac{1}{7} a^{13} - \frac{1}{14} a^{12} - \frac{3}{14} a^{11} - \frac{5}{14} a^{8} - \frac{5}{14} a^{7} - \frac{3}{7} a^{6} + \frac{3}{14} a^{5} + \frac{1}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{102719877955710068789761410291854334460158427914295593046689967014} a^{19} - \frac{110090507053862597851100014155528831390691315130108528304519371}{102719877955710068789761410291854334460158427914295593046689967014} a^{18} - \frac{13872573975360280326415545805001478357876014428934890688838138733}{102719877955710068789761410291854334460158427914295593046689967014} a^{17} - \frac{24607810659689778004337459658588207301375351335503797835245499239}{102719877955710068789761410291854334460158427914295593046689967014} a^{16} + \frac{93115371334771818875493740990855934460702300048421159646774443}{14674268279387152684251630041693476351451203987756513292384281002} a^{15} + \frac{1496799816620421094491877373762135861003101107519538606142122760}{51359938977855034394880705145927167230079213957147796523344983507} a^{14} + \frac{4766538410797455225219374352716113494617170193598734336272109077}{34239959318570022929920470097284778153386142638098531015563322338} a^{13} + \frac{20699059060994758477237701892636843948299125652639736504970956405}{102719877955710068789761410291854334460158427914295593046689967014} a^{12} - \frac{13036564868740606977588832006112740694828021469112047897058100927}{102719877955710068789761410291854334460158427914295593046689967014} a^{11} - \frac{947903191609549699851796973065991704915517222511045971737976110}{7337134139693576342125815020846738175725601993878256646192140501} a^{10} - \frac{21018180551467633426027496822674245879113566033550419089021238925}{102719877955710068789761410291854334460158427914295593046689967014} a^{9} - \frac{3698218052197372429828637693062536841806554278939747485695596429}{51359938977855034394880705145927167230079213957147796523344983507} a^{8} + \frac{9494499257576817842083394110280932155166414278246027699521748747}{34239959318570022929920470097284778153386142638098531015563322338} a^{7} - \frac{35016226000340090277816396319324157053139587946928500471543265847}{102719877955710068789761410291854334460158427914295593046689967014} a^{6} + \frac{40324313666705954415689880383197348837129671742921248453393210515}{102719877955710068789761410291854334460158427914295593046689967014} a^{5} - \frac{5704587525170076545933447641291396262309345504716613428874441385}{17119979659285011464960235048642389076693071319049265507781661169} a^{4} - \frac{43705649472202288771472545362741557757702265300579463833990769}{2445711379897858780708605006948912725241867331292752215397380167} a^{3} - \frac{13121676709169190764890915959943185329050934404554084437784877}{462702152953648958512438785098442948018731657271601770480585437} a^{2} - \frac{13949739945379864821356594380231405617118220444902965846757890583}{51359938977855034394880705145927167230079213957147796523344983507} a - \frac{19658426845391760744659696327008743751087994825636843203909197065}{51359938977855034394880705145927167230079213957147796523344983507}$

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:  $11$
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:  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

$C_2\times F_5$ (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

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 5.1.50000.1, 10.2.256389362500000000.1, 10.2.51277872500000000.1, 10.2.12500000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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}$
$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}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$