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Properties

Label 20.12.3530940612...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{23}\cdot 7^{10}$
Root discriminant $33.68$
Ramified primes $2, 5, 7$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4:C_5:C_4$ (as 20T88)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 0, -325, 50, 1475, -400, -2800, 550, 2375, 950, -2035, -700, 715, 90, 60, -10, 5, -10, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 15*x^18 - 10*x^17 + 5*x^16 - 10*x^15 + 60*x^14 + 90*x^13 + 715*x^12 - 700*x^11 - 2035*x^10 + 950*x^9 + 2375*x^8 + 550*x^7 - 2800*x^6 - 400*x^5 + 1475*x^4 + 50*x^3 - 325*x^2 + 25)
 
gp: K = bnfinit(x^20 - 15*x^18 - 10*x^17 + 5*x^16 - 10*x^15 + 60*x^14 + 90*x^13 + 715*x^12 - 700*x^11 - 2035*x^10 + 950*x^9 + 2375*x^8 + 550*x^7 - 2800*x^6 - 400*x^5 + 1475*x^4 + 50*x^3 - 325*x^2 + 25, 1)
 

Normalized defining polynomial

\( x^{20} - 15 x^{18} - 10 x^{17} + 5 x^{16} - 10 x^{15} + 60 x^{14} + 90 x^{13} + 715 x^{12} - 700 x^{11} - 2035 x^{10} + 950 x^{9} + 2375 x^{8} + 550 x^{7} - 2800 x^{6} - 400 x^{5} + 1475 x^{4} + 50 x^{3} - 325 x^{2} + 25 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3530940612500000000000000000000=2^{20}\cdot 5^{23}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.68$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10}$, $\frac{1}{5} a^{11}$, $\frac{1}{5} a^{12}$, $\frac{1}{5} a^{13}$, $\frac{1}{5} a^{14}$, $\frac{1}{5} a^{15}$, $\frac{1}{25} a^{16} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{6}$, $\frac{1}{725} a^{17} - \frac{8}{725} a^{16} + \frac{13}{145} a^{15} - \frac{12}{145} a^{14} - \frac{8}{145} a^{13} - \frac{12}{145} a^{12} - \frac{4}{145} a^{11} + \frac{4}{145} a^{10} - \frac{53}{145} a^{9} - \frac{72}{145} a^{8} - \frac{56}{145} a^{7} - \frac{47}{145} a^{6} - \frac{13}{29} a^{5} + \frac{6}{29} a^{4} + \frac{8}{29} a^{3} + \frac{14}{29} a^{2} + \frac{9}{29} a + \frac{8}{29}$, $\frac{1}{725} a^{18} + \frac{1}{725} a^{16} + \frac{1}{29} a^{15} + \frac{12}{145} a^{14} + \frac{11}{145} a^{13} - \frac{13}{145} a^{12} + \frac{1}{145} a^{11} + \frac{8}{145} a^{10} - \frac{61}{145} a^{9} - \frac{52}{145} a^{8} - \frac{12}{29} a^{7} - \frac{6}{145} a^{6} - \frac{11}{29} a^{5} - \frac{2}{29} a^{4} - \frac{9}{29} a^{3} + \frac{5}{29} a^{2} - \frac{7}{29} a + \frac{6}{29}$, $\frac{1}{117355996590015275} a^{19} - \frac{44919680690596}{117355996590015275} a^{18} + \frac{75338609950637}{117355996590015275} a^{17} - \frac{374699147405144}{23471199318003055} a^{16} + \frac{1433020550764121}{23471199318003055} a^{15} + \frac{2011075913105014}{23471199318003055} a^{14} + \frac{4637732439871}{572468276048855} a^{13} + \frac{151531470703206}{4694239863600611} a^{12} - \frac{2159703504429189}{23471199318003055} a^{11} - \frac{151853060646031}{23471199318003055} a^{10} + \frac{5447424727767112}{23471199318003055} a^{9} - \frac{10671267077686814}{23471199318003055} a^{8} + \frac{7741330053436648}{23471199318003055} a^{7} - \frac{1485686321045485}{4694239863600611} a^{6} + \frac{1137838222539556}{4694239863600611} a^{5} - \frac{2007957403077792}{4694239863600611} a^{4} + \frac{2192184580760098}{4694239863600611} a^{3} + \frac{2247484599556448}{4694239863600611} a^{2} - \frac{1816995385108146}{4694239863600611} a + \frac{2224619740090814}{4694239863600611}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $15$
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:  \( 85671329.9317 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_5:C_4$ (as 20T88):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 11 conjugacy class representatives for $C_2^4:C_5:C_4$
Character table for $C_2^4:C_5:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.2450000.1, 10.10.30012500000000.1

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
Degree 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
5Data not computed
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$