↑

Properties

Label 20.0.48807303526...9200.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{14}\cdot 5^{2}\cdot 17\cdot 389^{4}$
Root discriminant $19.25$
Ramified primes $2, 3, 5, 17, 389$
Class number $2$
Class group $[2]$
Galois group 20T925

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10, -30, -11, 110, -66, -124, 153, 36, -123, 12, 48, 12, -15, -36, 30, 4, -6, -2, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + x^18 - 2*x^17 - 6*x^16 + 4*x^15 + 30*x^14 - 36*x^13 - 15*x^12 + 12*x^11 + 48*x^10 + 12*x^9 - 123*x^8 + 36*x^7 + 153*x^6 - 124*x^5 - 66*x^4 + 110*x^3 - 11*x^2 - 30*x + 10)
 
gp: K = bnfinit(x^20 + x^18 - 2*x^17 - 6*x^16 + 4*x^15 + 30*x^14 - 36*x^13 - 15*x^12 + 12*x^11 + 48*x^10 + 12*x^9 - 123*x^8 + 36*x^7 + 153*x^6 - 124*x^5 - 66*x^4 + 110*x^3 - 11*x^2 - 30*x + 10, 1)
 

Normalized defining polynomial

\( x^{20} + x^{18} - 2 x^{17} - 6 x^{16} + 4 x^{15} + 30 x^{14} - 36 x^{13} - 15 x^{12} + 12 x^{11} + 48 x^{10} + 12 x^{9} - 123 x^{8} + 36 x^{7} + 153 x^{6} - 124 x^{5} - 66 x^{4} + 110 x^{3} - 11 x^{2} - 30 x + 10 \)

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:  \(48807303526718226903859200=2^{20}\cdot 3^{14}\cdot 5^{2}\cdot 17\cdot 389^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.25$
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:  $2, 3, 5, 17, 389$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{15} + \frac{1}{9} a^{13} + \frac{1}{3} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{8} - \frac{4}{9} a^{7} + \frac{1}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{9} a^{3} + \frac{1}{3} a^{2} - \frac{2}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{15} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{11} + \frac{2}{9} a^{10} + \frac{4}{9} a^{9} - \frac{1}{9} a^{8} - \frac{2}{9} a^{6} - \frac{1}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2} + \frac{2}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{4}{9} a^{11} - \frac{4}{9} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} + \frac{2}{9} a^{4} - \frac{4}{9} a^{2} - \frac{4}{9} a + \frac{1}{9}$, $\frac{1}{335998833678} a^{19} - \frac{4066994779}{335998833678} a^{18} - \frac{2124248102}{167999416839} a^{17} - \frac{7033294271}{167999416839} a^{16} + \frac{2710694774}{18666601871} a^{15} - \frac{557116730}{167999416839} a^{14} + \frac{1235894102}{167999416839} a^{13} + \frac{9507379331}{167999416839} a^{12} - \frac{33001016629}{335998833678} a^{11} + \frac{75950012641}{335998833678} a^{10} - \frac{136966443317}{335998833678} a^{9} + \frac{15357969661}{335998833678} a^{8} - \frac{68862207425}{167999416839} a^{7} + \frac{72971545375}{167999416839} a^{6} + \frac{46745495521}{111999611226} a^{5} + \frac{158256668633}{335998833678} a^{4} - \frac{42941743073}{335998833678} a^{3} - \frac{154420771789}{335998833678} a^{2} - \frac{25613905436}{167999416839} a + \frac{20251405136}{55999805613}$

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:  \( \frac{37237033575}{37333203742} a^{19} + \frac{18089270573}{37333203742} a^{18} + \frac{25182750708}{18666601871} a^{17} - \frac{26162952984}{18666601871} a^{16} - \frac{122034637410}{18666601871} a^{15} + \frac{8276337450}{18666601871} a^{14} + \frac{551906647461}{18666601871} a^{13} - \frac{389143659054}{18666601871} a^{12} - \frac{812329361505}{37333203742} a^{11} - \frac{156145214925}{37333203742} a^{10} + \frac{1738241536913}{37333203742} a^{9} + \frac{1283349259845}{37333203742} a^{8} - \frac{1850565962205}{18666601871} a^{7} - \frac{251477735085}{18666601871} a^{6} + \frac{4949042465637}{37333203742} a^{5} - \frac{1918927229103}{37333203742} a^{4} - \frac{2896954768023}{37333203742} a^{3} + \frac{2059653039849}{37333203742} a^{2} + \frac{303240108825}{18666601871} a - \frac{261786271292}{18666601871} \) (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:  \( 198372.566362 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T925:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 409600
The 190 conjugacy class representatives for t20n925 are not computed
Character table for t20n925 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 10.0.112960521216.1

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

Sibling fields

Degree 20 siblings: 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 R R $20$ $20$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
17.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
389Data not computed