↑

Properties

Label 20.0.14105549537...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{16}\cdot 5^{15}$
Root discriminant $22.78$
Ramified primes $2, 3, 5$
Class number $2$
Class group $[2]$
Galois group $F_5$ (as 20T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![320, -960, 2160, -2960, 6476, -7176, 10214, -7336, 8525, -5026, 4497, -2238, 1763, -688, 492, -152, 99, -22, 13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 13*x^18 - 22*x^17 + 99*x^16 - 152*x^15 + 492*x^14 - 688*x^13 + 1763*x^12 - 2238*x^11 + 4497*x^10 - 5026*x^9 + 8525*x^8 - 7336*x^7 + 10214*x^6 - 7176*x^5 + 6476*x^4 - 2960*x^3 + 2160*x^2 - 960*x + 320)
 
gp: K = bnfinit(x^20 - 2*x^19 + 13*x^18 - 22*x^17 + 99*x^16 - 152*x^15 + 492*x^14 - 688*x^13 + 1763*x^12 - 2238*x^11 + 4497*x^10 - 5026*x^9 + 8525*x^8 - 7336*x^7 + 10214*x^6 - 7176*x^5 + 6476*x^4 - 2960*x^3 + 2160*x^2 - 960*x + 320, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 13 x^{18} - 22 x^{17} + 99 x^{16} - 152 x^{15} + 492 x^{14} - 688 x^{13} + 1763 x^{12} - 2238 x^{11} + 4497 x^{10} - 5026 x^{9} + 8525 x^{8} - 7336 x^{7} + 10214 x^{6} - 7176 x^{5} + 6476 x^{4} - 2960 x^{3} + 2160 x^{2} - 960 x + 320 \)

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:  \(1410554953728000000000000000=2^{30}\cdot 3^{16}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.78$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{3}{8} a^{7} - \frac{3}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} - \frac{1}{2} a^{7} - \frac{3}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{16} a^{9} + \frac{3}{16} a^{8} + \frac{5}{16} a^{7} - \frac{1}{16} a^{6} - \frac{3}{8} a^{5} - \frac{3}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{80} a^{16} + \frac{1}{40} a^{15} - \frac{9}{80} a^{12} - \frac{19}{80} a^{10} - \frac{7}{40} a^{8} - \frac{3}{8} a^{7} - \frac{1}{16} a^{6} + \frac{7}{20} a^{5} - \frac{7}{40} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{1760} a^{17} + \frac{1}{176} a^{16} + \frac{13}{880} a^{15} + \frac{1}{176} a^{14} + \frac{101}{1760} a^{13} - \frac{31}{880} a^{12} - \frac{79}{1760} a^{11} - \frac{3}{40} a^{10} + \frac{1}{10} a^{9} - \frac{3}{440} a^{8} - \frac{159}{352} a^{7} - \frac{261}{880} a^{6} - \frac{65}{176} a^{5} - \frac{93}{440} a^{4} - \frac{35}{88} a^{3} + \frac{9}{44} a^{2} - \frac{3}{22} a + \frac{1}{11}$, $\frac{1}{8800} a^{18} - \frac{1}{1100} a^{16} + \frac{53}{2200} a^{15} - \frac{109}{8800} a^{14} - \frac{261}{4400} a^{13} - \frac{163}{8800} a^{12} - \frac{1}{4400} a^{11} + \frac{81}{400} a^{10} - \frac{391}{4400} a^{9} - \frac{389}{8800} a^{8} - \frac{1291}{4400} a^{7} + \frac{193}{880} a^{6} + \frac{949}{2200} a^{5} + \frac{1019}{2200} a^{4} + \frac{19}{220} a^{3} + \frac{4}{11} a^{2} - \frac{17}{55} a + \frac{12}{55}$, $\frac{1}{13289851948940679200} a^{19} + \frac{9496992900629}{13289851948940679200} a^{18} + \frac{62265950791741}{6644925974470339600} a^{17} - \frac{3187763800888269}{664492597447033960} a^{16} + \frac{347622666847707259}{13289851948940679200} a^{15} - \frac{130748419855852883}{13289851948940679200} a^{14} - \frac{5118693471237811}{13289851948940679200} a^{13} - \frac{755971937696475069}{13289851948940679200} a^{12} + \frac{149503710826005407}{6644925974470339600} a^{11} - \frac{270191747404635961}{3322462987235169800} a^{10} + \frac{3006232847138520773}{13289851948940679200} a^{9} + \frac{1464725927908066997}{13289851948940679200} a^{8} + \frac{3200844822635024251}{6644925974470339600} a^{7} - \frac{1103405263850650507}{6644925974470339600} a^{6} - \frac{179134060205253771}{664492597447033960} a^{5} - \frac{285250176821509219}{3322462987235169800} a^{4} + \frac{1257253881222943}{15102104487432590} a^{3} + \frac{38359799276830858}{83061574680879245} a^{2} - \frac{8920834791235606}{83061574680879245} a - \frac{13933462536204302}{83061574680879245}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 639468.319487 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.8000.2, 5.1.648000.1 x5, 10.2.2099520000000.5 x5

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

Sibling fields

Degree 5 sibling: 5.1.648000.1
Degree 10 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
3Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$