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Properties

Label 20.0.42268800445...5744.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 89^{8}$
Root discriminant $17.03$
Ramified primes $2, 89$
Class number $2$
Class group $[2]$
Galois group 20T226

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -24, 72, -136, 164, -84, -136, 428, -611, 484, 0, -648, 1134, -1242, 1032, -696, 381, -162, 50, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 50*x^18 - 162*x^17 + 381*x^16 - 696*x^15 + 1032*x^14 - 1242*x^13 + 1134*x^12 - 648*x^11 + 484*x^9 - 611*x^8 + 428*x^7 - 136*x^6 - 84*x^5 + 164*x^4 - 136*x^3 + 72*x^2 - 24*x + 4)
 
gp: K = bnfinit(x^20 - 10*x^19 + 50*x^18 - 162*x^17 + 381*x^16 - 696*x^15 + 1032*x^14 - 1242*x^13 + 1134*x^12 - 648*x^11 + 484*x^9 - 611*x^8 + 428*x^7 - 136*x^6 - 84*x^5 + 164*x^4 - 136*x^3 + 72*x^2 - 24*x + 4, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 50 x^{18} - 162 x^{17} + 381 x^{16} - 696 x^{15} + 1032 x^{14} - 1242 x^{13} + 1134 x^{12} - 648 x^{11} + 484 x^{9} - 611 x^{8} + 428 x^{7} - 136 x^{6} - 84 x^{5} + 164 x^{4} - 136 x^{3} + 72 x^{2} - 24 x + 4 \)

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:  \(4226880044572534053535744=2^{30}\cdot 89^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.03$
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:  $2, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{4} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{18} - \frac{1}{2} a^{15} - \frac{1}{4} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{1554383298904} a^{19} - \frac{6864650097}{777191649452} a^{18} - \frac{44487441741}{388595824726} a^{17} + \frac{8169423812}{194297912363} a^{16} + \frac{507125431207}{1554383298904} a^{15} + \frac{66195900970}{194297912363} a^{14} - \frac{276288245633}{777191649452} a^{13} - \frac{184800390649}{388595824726} a^{12} - \frac{6123867269}{388595824726} a^{11} + \frac{41207207462}{194297912363} a^{10} + \frac{104839430299}{388595824726} a^{9} - \frac{9997615768}{194297912363} a^{8} + \frac{178314392605}{1554383298904} a^{7} - \frac{177414228995}{388595824726} a^{6} + \frac{375518772685}{777191649452} a^{5} - \frac{67427372291}{777191649452} a^{4} + \frac{41845542015}{777191649452} a^{3} + \frac{129295742489}{388595824726} a^{2} + \frac{4923610683}{388595824726} a + \frac{62571449457}{388595824726}$

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:  \( \frac{771012064493}{777191649452} a^{19} - \frac{3608982085647}{388595824726} a^{18} + \frac{8503016068609}{194297912363} a^{17} - \frac{51913716246287}{388595824726} a^{16} + \frac{230431720943411}{777191649452} a^{15} - \frac{99606107761039}{194297912363} a^{14} + \frac{280382920973029}{388595824726} a^{13} - \frac{316206346176265}{388595824726} a^{12} + \frac{128797303891451}{194297912363} a^{11} - \frac{54556994151091}{194297912363} a^{10} - \frac{26282456722782}{194297912363} a^{9} + \frac{74727646526329}{194297912363} a^{8} - \frac{299353923183435}{777191649452} a^{7} + \frac{42302443066471}{194297912363} a^{6} - \frac{10056039403453}{388595824726} a^{5} - \frac{17390313661730}{194297912363} a^{4} + \frac{42986294189331}{388595824726} a^{3} - \frac{14411682416492}{194297912363} a^{2} + \frac{6275845837401}{194297912363} a - \frac{1433630625901}{194297912363} \) (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:  \( 33768.3641595 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T226:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1920
The 18 conjugacy class representatives for t20n226
Character table for t20n226

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.3.31684.1, 10.0.64248054784.1, 10.6.256992219136.1, 10.0.256992219136.1

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 30 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.12.18.61$x^{12} - 6 x^{10} + 2 x^{8} - 4 x^{7} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 8$$4$$3$$18$$C_2^2 \times A_4$$[2, 2, 2]^{6}$
$89$89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.6.4.1$x^{6} + 1513 x^{3} + 1710936$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
89.6.4.1$x^{6} + 1513 x^{3} + 1710936$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$