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Properties

Label 20.12.5739871043...3125.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{4}\cdot 5^{17}\cdot 23^{6}\cdot 89^{4}$
Root discriminant $30.76$
Ramified primes $3, 5, 23, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T802

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, 0, 180, -330, -595, 2727, -3155, -1000, 6575, -7210, 1949, 3640, -4875, 2820, -670, -272, 340, -165, 50, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 50*x^18 - 165*x^17 + 340*x^16 - 272*x^15 - 670*x^14 + 2820*x^13 - 4875*x^12 + 3640*x^11 + 1949*x^10 - 7210*x^9 + 6575*x^8 - 1000*x^7 - 3155*x^6 + 2727*x^5 - 595*x^4 - 330*x^3 + 180*x^2 - 9)
 
gp: K = bnfinit(x^20 - 10*x^19 + 50*x^18 - 165*x^17 + 340*x^16 - 272*x^15 - 670*x^14 + 2820*x^13 - 4875*x^12 + 3640*x^11 + 1949*x^10 - 7210*x^9 + 6575*x^8 - 1000*x^7 - 3155*x^6 + 2727*x^5 - 595*x^4 - 330*x^3 + 180*x^2 - 9, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 50 x^{18} - 165 x^{17} + 340 x^{16} - 272 x^{15} - 670 x^{14} + 2820 x^{13} - 4875 x^{12} + 3640 x^{11} + 1949 x^{10} - 7210 x^{9} + 6575 x^{8} - 1000 x^{7} - 3155 x^{6} + 2727 x^{5} - 595 x^{4} - 330 x^{3} + 180 x^{2} - 9 \)

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:  \(573987104314626441192626953125=3^{4}\cdot 5^{17}\cdot 23^{6}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.76$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{10} + \frac{2}{5} a^{5} - \frac{2}{5}$, $\frac{1}{375} a^{16} - \frac{8}{375} a^{15} - \frac{2}{25} a^{14} - \frac{1}{15} a^{13} - \frac{2}{75} a^{12} + \frac{7}{125} a^{11} + \frac{14}{125} a^{10} - \frac{3}{25} a^{9} - \frac{3}{25} a^{8} + \frac{8}{75} a^{7} + \frac{127}{375} a^{6} - \frac{2}{125} a^{5} + \frac{7}{15} a^{4} - \frac{31}{75} a^{3} - \frac{4}{15} a^{2} + \frac{6}{125} a - \frac{33}{125}$, $\frac{1}{375} a^{17} - \frac{19}{375} a^{15} + \frac{7}{75} a^{14} + \frac{1}{25} a^{13} + \frac{16}{375} a^{12} - \frac{6}{25} a^{11} - \frac{53}{125} a^{10} + \frac{3}{25} a^{9} - \frac{34}{75} a^{8} - \frac{1}{125} a^{7} + \frac{37}{75} a^{6} + \frac{52}{375} a^{5} - \frac{12}{25} a^{4} - \frac{13}{75} a^{3} - \frac{107}{375} a^{2} - \frac{2}{25} a - \frac{14}{125}$, $\frac{1}{653818125} a^{18} - \frac{3}{217939375} a^{17} - \frac{520708}{653818125} a^{16} + \frac{4165868}{653818125} a^{15} - \frac{4156262}{43587875} a^{14} - \frac{28786769}{653818125} a^{13} + \frac{4834302}{217939375} a^{12} + \frac{28156444}{217939375} a^{11} + \frac{57683846}{217939375} a^{10} - \frac{2133304}{130763625} a^{9} + \frac{92700469}{217939375} a^{8} - \frac{216446848}{653818125} a^{7} + \frac{148624234}{653818125} a^{6} + \frac{41120687}{217939375} a^{5} + \frac{54624206}{130763625} a^{4} - \frac{286720352}{653818125} a^{3} + \frac{91440736}{217939375} a^{2} + \frac{49036642}{217939375} a - \frac{5162937}{217939375}$, $\frac{1}{653818125} a^{19} - \frac{22643}{28426875} a^{17} - \frac{520504}{653818125} a^{16} - \frac{8283706}{217939375} a^{15} + \frac{63935986}{653818125} a^{14} + \frac{1129949}{43587875} a^{13} - \frac{15510588}{217939375} a^{12} + \frac{5976717}{217939375} a^{11} + \frac{239161072}{653818125} a^{10} + \frac{60700909}{217939375} a^{9} + \frac{12696838}{130763625} a^{8} - \frac{230233898}{653818125} a^{7} - \frac{79648986}{217939375} a^{6} + \frac{75743329}{653818125} a^{5} + \frac{209914543}{653818125} a^{4} + \frac{3172286}{43587875} a^{3} + \frac{87421516}{217939375} a^{2} + \frac{87463841}{217939375} a - \frac{46466433}{217939375}$

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:  \( 60471799.5436 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T802:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n802 are not computed
Character table for t20n802 is not computed

Intermediate fields

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

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 $20$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
5Data not computed
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89Data not computed