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Properties

Label 20.0.45173470609...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{15}\cdot 13^{10}$
Root discriminant $34.10$
Ramified primes $2, 5, 13$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![400, -1600, 6160, -3120, 4756, 8024, 4088, -1204, 6303, 724, 880, 240, 516, 40, 160, -4, 8, 4, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 8*x^18 + 4*x^17 + 8*x^16 - 4*x^15 + 160*x^14 + 40*x^13 + 516*x^12 + 240*x^11 + 880*x^10 + 724*x^9 + 6303*x^8 - 1204*x^7 + 4088*x^6 + 8024*x^5 + 4756*x^4 - 3120*x^3 + 6160*x^2 - 1600*x + 400)
 
gp: K = bnfinit(x^20 - 4*x^19 + 8*x^18 + 4*x^17 + 8*x^16 - 4*x^15 + 160*x^14 + 40*x^13 + 516*x^12 + 240*x^11 + 880*x^10 + 724*x^9 + 6303*x^8 - 1204*x^7 + 4088*x^6 + 8024*x^5 + 4756*x^4 - 3120*x^3 + 6160*x^2 - 1600*x + 400, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 8 x^{18} + 4 x^{17} + 8 x^{16} - 4 x^{15} + 160 x^{14} + 40 x^{13} + 516 x^{12} + 240 x^{11} + 880 x^{10} + 724 x^{9} + 6303 x^{8} - 1204 x^{7} + 4088 x^{6} + 8024 x^{5} + 4756 x^{4} - 3120 x^{3} + 6160 x^{2} - 1600 x + 400 \)

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:  \(4517347060908032000000000000000=2^{30}\cdot 5^{15}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.10$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7}$, $\frac{1}{20} a^{14} + \frac{3}{20} a^{13} - \frac{1}{4} a^{11} + \frac{1}{20} a^{10} + \frac{1}{20} a^{8} - \frac{1}{4} a^{7} - \frac{1}{5} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{20} a^{15} + \frac{1}{20} a^{13} - \frac{1}{4} a^{12} - \frac{1}{5} a^{11} - \frac{3}{20} a^{10} + \frac{1}{20} a^{9} + \frac{1}{10} a^{8} + \frac{1}{20} a^{7} + \frac{7}{20} a^{6} - \frac{3}{20} a^{4} + \frac{1}{10} a^{2}$, $\frac{1}{120} a^{16} + \frac{1}{60} a^{15} + \frac{1}{5} a^{13} + \frac{1}{20} a^{12} - \frac{13}{60} a^{11} + \frac{1}{30} a^{10} + \frac{7}{60} a^{9} - \frac{13}{60} a^{8} - \frac{3}{10} a^{7} - \frac{7}{20} a^{6} - \frac{3}{20} a^{5} + \frac{19}{120} a^{4} + \frac{1}{6} a^{3} + \frac{3}{20} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{120} a^{17} + \frac{1}{60} a^{15} + \frac{1}{10} a^{13} - \frac{1}{15} a^{12} - \frac{7}{30} a^{11} + \frac{1}{5} a^{10} + \frac{1}{10} a^{9} + \frac{1}{30} a^{8} + \frac{3}{10} a^{7} + \frac{1}{5} a^{6} - \frac{1}{24} a^{5} + \frac{1}{5} a^{4} - \frac{1}{12} a^{3} - \frac{1}{15} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{1200} a^{18} + \frac{1}{600} a^{17} - \frac{1}{300} a^{16} - \frac{1}{150} a^{15} - \frac{1}{50} a^{14} - \frac{37}{150} a^{13} + \frac{7}{30} a^{12} - \frac{29}{300} a^{11} + \frac{1}{5} a^{10} + \frac{4}{75} a^{9} + \frac{11}{300} a^{8} - \frac{1}{25} a^{7} - \frac{401}{1200} a^{6} - \frac{59}{600} a^{5} + \frac{14}{75} a^{4} + \frac{7}{60} a^{3} + \frac{11}{30} a^{2} + \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{86035151345091428158233141349200} a^{19} + \frac{98596396825135353258824553}{7169595945424285679852761779100} a^{18} - \frac{54020306545657945797705337729}{21508787836272857039558285337300} a^{17} + \frac{80965711322210253900750771259}{21508787836272857039558285337300} a^{16} - \frac{183353958038539911668126680097}{10754393918136428519779142668650} a^{15} - \frac{157399974222374570254603480729}{10754393918136428519779142668650} a^{14} - \frac{520610866286216897653105554079}{5377196959068214259889571334325} a^{13} - \frac{1207096454116070460661848404273}{7169595945424285679852761779100} a^{12} - \frac{2112403584325994661010724195203}{10754393918136428519779142668650} a^{11} + \frac{1385734528505397499945415010943}{10754393918136428519779142668650} a^{10} + \frac{69933795852622999729215823883}{860351513450914281582331413492} a^{9} - \frac{1009952359096389969119079630077}{5377196959068214259889571334325} a^{8} - \frac{37577205456249847803213050184473}{86035151345091428158233141349200} a^{7} - \frac{1160826565129078224809923128663}{3584797972712142839926380889550} a^{6} + \frac{668230910332652711041277728009}{10754393918136428519779142668650} a^{5} - \frac{2630810762214084865838724363713}{10754393918136428519779142668650} a^{4} - \frac{393433880889895950523868677208}{1075439391813642851977914266865} a^{3} + \frac{204470189170946112325890948127}{1075439391813642851977914266865} a^{2} + \frac{150902316574213000978731585431}{430175756725457140791165706746} a + \frac{101176899732756210582031220330}{215087878362728570395582853373}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 14979999.4048 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), 4.0.1352000.1, 5.1.1352000.1 x5, 10.2.9139520000000.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.1352000.1
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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$$[3]^{2}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$