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Properties

Label 20.4.56567708040...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 3^{10}\cdot 5^{22}\cdot 19^{10}$
Root discriminant $77.20$
Ramified primes $2, 3, 5, 19$
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![575052033079, 3487818320, -347034273315, -552975930, 96104734755, -8822742, -16128380640, 7693290, 1818277605, -689170, -143852455, 26230, 8082595, -310, -318340, -4, 8415, 0, -135, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 135*x^18 + 8415*x^16 - 4*x^15 - 318340*x^14 - 310*x^13 + 8082595*x^12 + 26230*x^11 - 143852455*x^10 - 689170*x^9 + 1818277605*x^8 + 7693290*x^7 - 16128380640*x^6 - 8822742*x^5 + 96104734755*x^4 - 552975930*x^3 - 347034273315*x^2 + 3487818320*x + 575052033079)
 
gp: K = bnfinit(x^20 - 135*x^18 + 8415*x^16 - 4*x^15 - 318340*x^14 - 310*x^13 + 8082595*x^12 + 26230*x^11 - 143852455*x^10 - 689170*x^9 + 1818277605*x^8 + 7693290*x^7 - 16128380640*x^6 - 8822742*x^5 + 96104734755*x^4 - 552975930*x^3 - 347034273315*x^2 + 3487818320*x + 575052033079, 1)
 

Normalized defining polynomial

\( x^{20} - 135 x^{18} + 8415 x^{16} - 4 x^{15} - 318340 x^{14} - 310 x^{13} + 8082595 x^{12} + 26230 x^{11} - 143852455 x^{10} - 689170 x^{9} + 1818277605 x^{8} + 7693290 x^{7} - 16128380640 x^{6} - 8822742 x^{5} + 96104734755 x^{4} - 552975930 x^{3} - 347034273315 x^{2} + 3487818320 x + 575052033079 \)

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:  \(56567708040139257656250000000000000000=2^{16}\cdot 3^{10}\cdot 5^{22}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.20$
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:  $2, 3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} - \frac{1}{7} a^{16} - \frac{1}{7} a^{15} - \frac{3}{14} a^{14} + \frac{1}{7} a^{13} - \frac{1}{14} a^{12} - \frac{3}{14} a^{11} - \frac{5}{14} a^{8} - \frac{5}{14} a^{7} + \frac{1}{14} a^{6} + \frac{3}{14} a^{5} - \frac{5}{14} a^{4} - \frac{3}{14} a^{3} - \frac{1}{14} a^{2} + \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{509260951920653442245812370250118978601054308990935241167842506702723515849902} a^{19} - \frac{5903150414097845310812789377016582511852759487458509283054309501258028474346}{254630475960326721122906185125059489300527154495467620583921253351361757924951} a^{18} + \frac{10400031921597919623158147211891781861503034475886805556532719981055561719507}{509260951920653442245812370250118978601054308990935241167842506702723515849902} a^{17} + \frac{6866800356026692430943933140796793819261322945808356497756510890495352241341}{72751564560093348892258910035731282657293472712990748738263215243246216549986} a^{16} + \frac{9401054879882785172978388526348242521867485016199448209656361901931558818083}{254630475960326721122906185125059489300527154495467620583921253351361757924951} a^{15} - \frac{23764668837705951424701004578072880533971803972675471919651590300338285342844}{254630475960326721122906185125059489300527154495467620583921253351361757924951} a^{14} - \frac{55653541270682117954533086251865960170755346457442819549977404401576936912541}{509260951920653442245812370250118978601054308990935241167842506702723515849902} a^{13} + \frac{21740801439584407197712550112236750888022887055724805490021573260105882362561}{254630475960326721122906185125059489300527154495467620583921253351361757924951} a^{12} + \frac{28476823274391865419645507601472229975854858454985137789610145860731016172381}{509260951920653442245812370250118978601054308990935241167842506702723515849902} a^{11} - \frac{10312814275393500684563531439359574963760890418838639593055821550456987973799}{72751564560093348892258910035731282657293472712990748738263215243246216549986} a^{10} - \frac{113871772867774595675586880638995300886560573697862016683346693749183595053019}{254630475960326721122906185125059489300527154495467620583921253351361757924951} a^{9} - \frac{9719626107285889029098671296379469463464974103063866274838451059640512713671}{72751564560093348892258910035731282657293472712990748738263215243246216549986} a^{8} + \frac{97800113038688886605363522648018439953118287422265774725189966519714618523371}{254630475960326721122906185125059489300527154495467620583921253351361757924951} a^{7} - \frac{205139407065676945229163309505005426784652583199553142255230396886256759572803}{509260951920653442245812370250118978601054308990935241167842506702723515849902} a^{6} - \frac{21158237399046941582000343678574216872554936615554063177925823013041876696017}{509260951920653442245812370250118978601054308990935241167842506702723515849902} a^{5} - \frac{5341273801508893087223800595517682234752308057995274661540434120968189299929}{254630475960326721122906185125059489300527154495467620583921253351361757924951} a^{4} - \frac{86169798139345777691989786932979623964833524901088868798905740647286170660063}{509260951920653442245812370250118978601054308990935241167842506702723515849902} a^{3} + \frac{736817417003943925000358398411849922705666996890962478974886192366659253255}{72751564560093348892258910035731282657293472712990748738263215243246216549986} a^{2} - \frac{89885043746375602003939852940251609411272420575826012412095569620046270132401}{509260951920653442245812370250118978601054308990935241167842506702723515849902} a - \frac{40560409922838111743568858598769045538282366310506606428821153578723265259813}{509260951920653442245812370250118978601054308990935241167842506702723515849902}$

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

\(\Q(\sqrt{57}) \), \(\Q(\sqrt{285}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{57})\), 5.1.50000.1, 10.2.7521150712500000000.1, 10.2.1504230142500000000.1, 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$$[\ ]_{2}^{4}$
5Data not computed
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$