Properties

Label 20.2.120...008.1
Degree $20$
Signature $[2, 9]$
Discriminant $-1.202\times 10^{24}$
Root discriminant $16.00$
Ramified primes $2, 47$
Class number $1$
Class group trivial
Galois group $C_5:D_4$ (as 20T7)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - x^18 + 4*x^17 + 12*x^16 + 4*x^15 - 20*x^14 - 28*x^13 - 68*x^12 - 24*x^11 + 70*x^10 + 170*x^9 + 249*x^8 + 258*x^7 + 271*x^6 + 184*x^5 + 126*x^4 + 70*x^3 + 34*x^2 + 8*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - x^18 + 4*x^17 + 12*x^16 + 4*x^15 - 20*x^14 - 28*x^13 - 68*x^12 - 24*x^11 + 70*x^10 + 170*x^9 + 249*x^8 + 258*x^7 + 271*x^6 + 184*x^5 + 126*x^4 + 70*x^3 + 34*x^2 + 8*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 8, 34, 70, 126, 184, 271, 258, 249, 170, 70, -24, -68, -28, -20, 4, 12, 4, -1, -2, 1]);
 

\(x^{20} - 2 x^{19} - x^{18} + 4 x^{17} + 12 x^{16} + 4 x^{15} - 20 x^{14} - 28 x^{13} - 68 x^{12} - 24 x^{11} + 70 x^{10} + 170 x^{9} + 249 x^{8} + 258 x^{7} + 271 x^{6} + 184 x^{5} + 126 x^{4} + 70 x^{3} + 34 x^{2} + 8 x + 1\)  Toggle raw display

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-1201657195483347978027008\)\(\medspace = -\,2^{30}\cdot 47^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $16.00$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 47$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{4831} a^{18} + \frac{1313}{4831} a^{17} - \frac{1603}{4831} a^{16} + \frac{1245}{4831} a^{15} + \frac{967}{4831} a^{14} - \frac{2415}{4831} a^{13} + \frac{229}{4831} a^{12} - \frac{180}{4831} a^{11} - \frac{1642}{4831} a^{10} - \frac{2089}{4831} a^{9} + \frac{934}{4831} a^{8} - \frac{1461}{4831} a^{7} - \frac{506}{4831} a^{6} - \frac{632}{4831} a^{5} - \frac{1167}{4831} a^{4} + \frac{875}{4831} a^{3} - \frac{360}{4831} a^{2} - \frac{1633}{4831} a - \frac{2150}{4831}$, $\frac{1}{20604224748798577} a^{19} - \frac{1557362169739}{20604224748798577} a^{18} + \frac{8875788818054124}{20604224748798577} a^{17} + \frac{1084017166776655}{20604224748798577} a^{16} + \frac{937834589060012}{20604224748798577} a^{15} + \frac{3868289359814656}{20604224748798577} a^{14} - \frac{9150561619379025}{20604224748798577} a^{13} + \frac{8872297969835316}{20604224748798577} a^{12} - \frac{8859682837760490}{20604224748798577} a^{11} + \frac{41688562597430}{20604224748798577} a^{10} - \frac{1091119053598161}{20604224748798577} a^{9} - \frac{4252094152527717}{20604224748798577} a^{8} + \frac{4493754291608698}{20604224748798577} a^{7} + \frac{6823814776852366}{20604224748798577} a^{6} - \frac{3704734530023037}{20604224748798577} a^{5} - \frac{138043565413421}{664652411251567} a^{4} + \frac{8797713987635912}{20604224748798577} a^{3} - \frac{3128969813136656}{20604224748798577} a^{2} - \frac{2999343886386005}{20604224748798577} a + \frac{22144199823811}{20604224748798577}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $10$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 6974.62635242 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{9}\cdot 6974.62635242 \cdot 1}{2\sqrt{1201657195483347978027008}}\approx 0.194213520866$

Galois group

$C_5:D_4$ (as 20T7):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 40
The 13 conjugacy class representatives for $C_5:D_4$
Character table for $C_5:D_4$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.2.3008.1, 5.1.2209.1, 10.2.159897387008.2

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

Sibling fields

Galois closure: Deg 40
Degree 20 sibling: 20.0.1723568365103679045632.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$