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Properties

Label 20.12.2170083570...0625.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{4}\cdot 5^{16}\cdot 23^{4}\cdot 89^{4}$
Root discriminant $20.74$
Ramified primes $3, 5, 23, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T279

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 0, -35, -45, 66, 175, -5, -280, -75, 297, 75, -280, 5, 175, -66, -45, 35, 0, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 35*x^17 - 45*x^16 - 66*x^15 + 175*x^14 + 5*x^13 - 280*x^12 + 75*x^11 + 297*x^10 - 75*x^9 - 280*x^8 - 5*x^7 + 175*x^6 + 66*x^5 - 45*x^4 - 35*x^3 + 5*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 35*x^17 - 45*x^16 - 66*x^15 + 175*x^14 + 5*x^13 - 280*x^12 + 75*x^11 + 297*x^10 - 75*x^9 - 280*x^8 - 5*x^7 + 175*x^6 + 66*x^5 - 45*x^4 - 35*x^3 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 35 x^{17} - 45 x^{16} - 66 x^{15} + 175 x^{14} + 5 x^{13} - 280 x^{12} + 75 x^{11} + 297 x^{10} - 75 x^{9} - 280 x^{8} - 5 x^{7} + 175 x^{6} + 66 x^{5} - 45 x^{4} - 35 x^{3} + 5 x + 1 \)

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:  \(217008357018762359619140625=3^{4}\cdot 5^{16}\cdot 23^{4}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.74$
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:  $3, 5, 23, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{10} + \frac{2}{5} a^{8} - \frac{1}{5} a^{5} - \frac{1}{5} a^{3} - \frac{2}{5}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{11} + \frac{2}{5} a^{9} - \frac{1}{5} a^{6} - \frac{1}{5} a^{4} - \frac{2}{5} a$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{15} a^{16} - \frac{1}{15} a^{15} + \frac{1}{15} a^{11} + \frac{2}{15} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{2}{15} a^{6} - \frac{1}{15} a^{5} + \frac{1}{15} a + \frac{1}{15}$, $\frac{1}{15} a^{17} - \frac{1}{15} a^{15} + \frac{1}{15} a^{12} + \frac{1}{5} a^{11} + \frac{7}{15} a^{10} - \frac{1}{3} a^{9} - \frac{1}{5} a^{7} + \frac{1}{15} a^{6} - \frac{1}{15} a^{5} + \frac{1}{15} a^{2} + \frac{2}{15} a + \frac{1}{15}$, $\frac{1}{315} a^{18} + \frac{2}{105} a^{17} + \frac{4}{315} a^{16} - \frac{4}{63} a^{15} + \frac{4}{105} a^{14} + \frac{4}{315} a^{13} - \frac{1}{15} a^{12} + \frac{2}{21} a^{11} - \frac{13}{315} a^{10} + \frac{109}{315} a^{9} - \frac{92}{315} a^{8} + \frac{31}{105} a^{7} + \frac{109}{315} a^{5} - \frac{4}{105} a^{4} - \frac{62}{315} a^{3} + \frac{143}{315} a^{2} + \frac{17}{35} a + \frac{83}{315}$, $\frac{1}{408555} a^{19} + \frac{31}{408555} a^{18} - \frac{767}{58365} a^{17} + \frac{12974}{408555} a^{16} + \frac{32524}{408555} a^{15} - \frac{6878}{408555} a^{14} + \frac{1591}{408555} a^{13} + \frac{10447}{136185} a^{12} - \frac{124906}{408555} a^{11} + \frac{43286}{136185} a^{10} + \frac{183674}{408555} a^{9} + \frac{22132}{408555} a^{8} + \frac{18603}{45395} a^{7} + \frac{194758}{408555} a^{6} - \frac{97394}{408555} a^{5} + \frac{23540}{81711} a^{4} - \frac{4639}{19455} a^{3} + \frac{163391}{408555} a^{2} - \frac{111361}{408555} a - \frac{3365}{81711}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 798245.370635 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T279:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n279
Character table for t20n279 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.767625.1, 10.10.2946240703125.1, 10.6.2946240703125.1, 10.6.14731203515625.1

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 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$89$89.3.0.1$x^{3} - x + 7$$1$$3$$0$$C_3$$[\ ]^{3}$
89.3.0.1$x^{3} - x + 7$$1$$3$$0$$C_3$$[\ ]^{3}$
89.3.0.1$x^{3} - x + 7$$1$$3$$0$$C_3$$[\ ]^{3}$
89.3.0.1$x^{3} - x + 7$$1$$3$$0$$C_3$$[\ ]^{3}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$