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Properties

Label 20.0.33626538312...2249.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 7^{10}\cdot 17^{10}$
Root discriminant $18.89$
Ramified primes $3, 7, 17$
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![49, 147, 14, -399, 354, -315, 156, 546, -1049, 537, 727, -1689, 1648, -939, 296, -18, -18, -6, 13, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 13*x^18 - 6*x^17 - 18*x^16 - 18*x^15 + 296*x^14 - 939*x^13 + 1648*x^12 - 1689*x^11 + 727*x^10 + 537*x^9 - 1049*x^8 + 546*x^7 + 156*x^6 - 315*x^5 + 354*x^4 - 399*x^3 + 14*x^2 + 147*x + 49)
 
gp: K = bnfinit(x^20 - 6*x^19 + 13*x^18 - 6*x^17 - 18*x^16 - 18*x^15 + 296*x^14 - 939*x^13 + 1648*x^12 - 1689*x^11 + 727*x^10 + 537*x^9 - 1049*x^8 + 546*x^7 + 156*x^6 - 315*x^5 + 354*x^4 - 399*x^3 + 14*x^2 + 147*x + 49, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 13 x^{18} - 6 x^{17} - 18 x^{16} - 18 x^{15} + 296 x^{14} - 939 x^{13} + 1648 x^{12} - 1689 x^{11} + 727 x^{10} + 537 x^{9} - 1049 x^{8} + 546 x^{7} + 156 x^{6} - 315 x^{5} + 354 x^{4} - 399 x^{3} + 14 x^{2} + 147 x + 49 \)

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:  \(33626538312268515533112249=3^{10}\cdot 7^{10}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.89$
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:  $3, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} + \frac{3}{7} a^{9} - \frac{2}{7} a^{7} - \frac{2}{7} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{3} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{11} - \frac{2}{7} a^{9} - \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2}$, $\frac{1}{7} a^{12} - \frac{3}{7} a^{9} - \frac{3}{7} a^{8} - \frac{3}{7} a^{7} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{3}{7} a^{4} - \frac{2}{7} a^{3} + \frac{2}{7} a^{2}$, $\frac{1}{7} a^{13} - \frac{1}{7} a^{9} - \frac{3}{7} a^{8} - \frac{2}{7} a^{7} + \frac{2}{7} a^{6} + \frac{2}{7} a^{5} - \frac{2}{7} a^{4} + \frac{3}{7} a^{2}$, $\frac{1}{7} a^{14} - \frac{2}{7} a^{8} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{15} - \frac{2}{7} a^{9} + \frac{1}{7} a^{3}$, $\frac{1}{77} a^{16} - \frac{1}{77} a^{15} + \frac{4}{77} a^{14} - \frac{3}{77} a^{12} + \frac{3}{77} a^{11} + \frac{1}{77} a^{10} + \frac{3}{11} a^{9} - \frac{5}{77} a^{8} - \frac{6}{77} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{38}{77} a^{4} - \frac{30}{77} a^{3} - \frac{29}{77} a^{2} + \frac{2}{11} a + \frac{2}{11}$, $\frac{1}{539} a^{17} + \frac{1}{539} a^{16} - \frac{31}{539} a^{15} + \frac{8}{539} a^{14} - \frac{3}{539} a^{13} + \frac{30}{539} a^{12} - \frac{4}{539} a^{11} - \frac{32}{539} a^{10} + \frac{92}{539} a^{9} - \frac{16}{539} a^{8} + \frac{10}{539} a^{7} + \frac{8}{49} a^{6} - \frac{16}{539} a^{5} - \frac{205}{539} a^{4} + \frac{36}{77} a^{3} - \frac{1}{7} a^{2} + \frac{4}{11} a - \frac{1}{11}$, $\frac{1}{5929} a^{18} + \frac{2}{5929} a^{17} + \frac{26}{5929} a^{16} + \frac{75}{5929} a^{15} + \frac{383}{5929} a^{14} - \frac{204}{5929} a^{13} - \frac{65}{5929} a^{12} + \frac{3}{49} a^{11} + \frac{270}{5929} a^{10} - \frac{134}{5929} a^{9} + \frac{93}{539} a^{8} + \frac{318}{847} a^{7} + \frac{2921}{5929} a^{6} + \frac{2474}{5929} a^{5} + \frac{2231}{5929} a^{4} - \frac{48}{121} a^{3} + \frac{192}{847} a^{2} - \frac{25}{121} a - \frac{40}{121}$, $\frac{1}{3706599769103} a^{19} + \frac{260748634}{3706599769103} a^{18} + \frac{2214247316}{3706599769103} a^{17} + \frac{5451986322}{3706599769103} a^{16} - \frac{219144348229}{3706599769103} a^{15} + \frac{33197503803}{529514252729} a^{14} - \frac{124140558547}{3706599769103} a^{13} + \frac{146853999469}{3706599769103} a^{12} - \frac{10530503131}{3706599769103} a^{11} + \frac{19275321801}{529514252729} a^{10} + \frac{1308627268661}{3706599769103} a^{9} + \frac{224293790841}{529514252729} a^{8} + \frac{847153867680}{3706599769103} a^{7} + \frac{261859526087}{3706599769103} a^{6} + \frac{452711529157}{3706599769103} a^{5} - \frac{95149250983}{336963615373} a^{4} + \frac{223729878833}{529514252729} a^{3} - \frac{141130155116}{529514252729} a^{2} - \frac{23378604050}{75644893247} a - \frac{19969904027}{75644893247}$

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:  \( \frac{574916563}{30633055943} a^{19} - \frac{2954859485}{30633055943} a^{18} + \frac{5347822169}{30633055943} a^{17} - \frac{1001458021}{30633055943} a^{16} - \frac{7656267522}{30633055943} a^{15} - \frac{2345627673}{4376150849} a^{14} + \frac{149427817232}{30633055943} a^{13} - \frac{425142505990}{30633055943} a^{12} + \frac{692612131753}{30633055943} a^{11} - \frac{95582213596}{4376150849} a^{10} + \frac{278158007802}{30633055943} a^{9} + \frac{28263712265}{4376150849} a^{8} - \frac{379184265344}{30633055943} a^{7} + \frac{203321325164}{30633055943} a^{6} + \frac{5781911775}{30633055943} a^{5} - \frac{74451936670}{30633055943} a^{4} + \frac{21946658744}{4376150849} a^{3} - \frac{19749481265}{4376150849} a^{2} + \frac{330991204}{625164407} a + \frac{597837970}{625164407} \) (order $6$)
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:  \( 107340.04077 \)
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

\(\Q(\sqrt{-119}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{357}) \), \(\Q(\sqrt{-3}, \sqrt{-119})\), 5.1.14161.1 x5, 10.0.23863536599.2, 10.0.48729742803.1 x5, 10.2.5798839393557.1 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$