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Properties

Label 20.0.37493871612...7424.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 11^{10}\cdot 13^{10}$
Root discriminant $23.92$
Ramified primes $2, 11, 13$
Class number $2$
Class group $[2]$
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, 0, -1925, 0, 5299, 0, -2480, 0, 4367, 0, -1557, 0, 744, 0, -233, 0, 60, 0, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 12*x^18 + 60*x^16 - 233*x^14 + 744*x^12 - 1557*x^10 + 4367*x^8 - 2480*x^6 + 5299*x^4 - 1925*x^2 + 625)
 
gp: K = bnfinit(x^20 - 12*x^18 + 60*x^16 - 233*x^14 + 744*x^12 - 1557*x^10 + 4367*x^8 - 2480*x^6 + 5299*x^4 - 1925*x^2 + 625, 1)
 

Normalized defining polynomial

\( x^{20} - 12 x^{18} + 60 x^{16} - 233 x^{14} + 744 x^{12} - 1557 x^{10} + 4367 x^{8} - 2480 x^{6} + 5299 x^{4} - 1925 x^{2} + 625 \)

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:  \(3749387161243233107049447424=2^{20}\cdot 11^{10}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.92$
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:  $2, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} - \frac{5}{11} a^{10} - \frac{3}{11} a^{8} + \frac{3}{11} a^{6} - \frac{1}{11} a^{4} - \frac{3}{11} a^{2} - \frac{2}{11}$, $\frac{1}{11} a^{13} - \frac{5}{11} a^{11} - \frac{3}{11} a^{9} + \frac{3}{11} a^{7} - \frac{1}{11} a^{5} - \frac{3}{11} a^{3} - \frac{2}{11} a$, $\frac{1}{11} a^{14} + \frac{5}{11} a^{10} - \frac{1}{11} a^{8} + \frac{3}{11} a^{6} + \frac{3}{11} a^{4} + \frac{5}{11} a^{2} + \frac{1}{11}$, $\frac{1}{55} a^{15} + \frac{1}{55} a^{13} - \frac{2}{5} a^{11} - \frac{4}{55} a^{9} + \frac{17}{55} a^{7} + \frac{24}{55} a^{5} + \frac{24}{55} a^{3} - \frac{23}{55} a$, $\frac{1}{7865} a^{16} + \frac{151}{7865} a^{14} + \frac{263}{7865} a^{12} - \frac{1779}{7865} a^{10} - \frac{218}{7865} a^{8} - \frac{2906}{7865} a^{6} - \frac{636}{7865} a^{4} - \frac{2878}{7865} a^{2} - \frac{106}{1573}$, $\frac{1}{7865} a^{17} + \frac{8}{7865} a^{15} + \frac{24}{1573} a^{13} + \frac{1367}{7865} a^{11} + \frac{354}{7865} a^{9} + \frac{2528}{7865} a^{7} + \frac{3797}{7865} a^{5} + \frac{311}{1573} a^{3} + \frac{2759}{7865} a$, $\frac{1}{21815787211075} a^{18} - \frac{1056285232}{21815787211075} a^{16} + \frac{181677981493}{4363157442215} a^{14} - \frac{449050379468}{21815787211075} a^{12} + \frac{6068802459274}{21815787211075} a^{10} + \frac{244261553873}{1983253382825} a^{8} - \frac{536925184058}{1983253382825} a^{6} + \frac{842302882288}{4363157442215} a^{4} + \frac{739071159334}{21815787211075} a^{2} + \frac{416637745021}{872631488443}$, $\frac{1}{109078936055375} a^{19} - \frac{3830066187}{109078936055375} a^{17} + \frac{97909796652}{21815787211075} a^{15} - \frac{1178554770633}{109078936055375} a^{13} - \frac{32628215643931}{109078936055375} a^{11} - \frac{1684020533662}{9916266914125} a^{9} + \frac{2179110694697}{9916266914125} a^{7} + \frac{1195127819764}{21815787211075} a^{5} + \frac{8722012747824}{109078936055375} a^{3} - \frac{2142452564062}{4363157442215} a$

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:  \( -\frac{79306}{67049125} a^{19} + \frac{70044}{5157625} a^{17} - \frac{850587}{13409825} a^{15} + \frac{15872898}{67049125} a^{13} - \frac{3754928}{5157625} a^{11} + \frac{8266672}{6095375} a^{9} - \frac{25276457}{6095375} a^{7} + \frac{235882}{1031525} a^{5} - \frac{302383469}{67049125} a^{3} - \frac{118821}{2681965} a \) (order $4$)
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:  \( 512830.066318 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-143}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{143}) \), \(\Q(i, \sqrt{143})\), 5.1.20449.1 x5, 10.0.59797108943.2, 10.0.428197479424.4 x5, 10.2.61232239557632.1 x5

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

Sibling fields

Degree 10 siblings: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$