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Properties

Label 20.0.46098141783...3549.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 61^{7}\cdot 397^{4}$
Root discriminant $24.16$
Ramified primes $3, 61, 397$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T466

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![109, -971, 3985, -9840, 16541, -20890, 21834, -20005, 15833, -10568, 6255, -3447, 1502, -345, 18, -15, -16, 32, -10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 10*x^18 + 32*x^17 - 16*x^16 - 15*x^15 + 18*x^14 - 345*x^13 + 1502*x^12 - 3447*x^11 + 6255*x^10 - 10568*x^9 + 15833*x^8 - 20005*x^7 + 21834*x^6 - 20890*x^5 + 16541*x^4 - 9840*x^3 + 3985*x^2 - 971*x + 109)
 
gp: K = bnfinit(x^20 - 2*x^19 - 10*x^18 + 32*x^17 - 16*x^16 - 15*x^15 + 18*x^14 - 345*x^13 + 1502*x^12 - 3447*x^11 + 6255*x^10 - 10568*x^9 + 15833*x^8 - 20005*x^7 + 21834*x^6 - 20890*x^5 + 16541*x^4 - 9840*x^3 + 3985*x^2 - 971*x + 109, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 10 x^{18} + 32 x^{17} - 16 x^{16} - 15 x^{15} + 18 x^{14} - 345 x^{13} + 1502 x^{12} - 3447 x^{11} + 6255 x^{10} - 10568 x^{9} + 15833 x^{8} - 20005 x^{7} + 21834 x^{6} - 20890 x^{5} + 16541 x^{4} - 9840 x^{3} + 3985 x^{2} - 971 x + 109 \)

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:  \(4609814178311274644985033549=3^{10}\cdot 61^{7}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.16$
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:  $3, 61, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{676695340331608719} a^{19} - \frac{40444860968468032}{676695340331608719} a^{18} - \frac{84893219618999557}{676695340331608719} a^{17} + \frac{2962691728627257}{20505919403988143} a^{16} + \frac{13801586877008420}{676695340331608719} a^{15} + \frac{96339022026284579}{676695340331608719} a^{14} - \frac{49076773630334830}{676695340331608719} a^{13} + \frac{910245502607448}{20505919403988143} a^{12} - \frac{14582819295496714}{225565113443869573} a^{11} + \frac{103181813165325403}{676695340331608719} a^{10} + \frac{172792940523061193}{676695340331608719} a^{9} - \frac{68156859077158075}{676695340331608719} a^{8} + \frac{52628149358307064}{676695340331608719} a^{7} + \frac{12467768356391840}{225565113443869573} a^{6} - \frac{4775467300867463}{225565113443869573} a^{5} - \frac{117792213241984223}{676695340331608719} a^{4} + \frac{56874851160200548}{225565113443869573} a^{3} - \frac{96755667168976474}{676695340331608719} a^{2} - \frac{105374994874322358}{225565113443869573} a + \frac{128129181524320574}{676695340331608719}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{37584503431}{172683145129} a^{19} + \frac{136157001260}{518049435387} a^{18} + \frac{1223084988244}{518049435387} a^{17} - \frac{79947290703}{15698467739} a^{16} - \frac{148609660796}{518049435387} a^{15} + \frac{1450915295812}{518049435387} a^{14} - \frac{355880455490}{172683145129} a^{13} + \frac{1150245074799}{15698467739} a^{12} - \frac{139464041570263}{518049435387} a^{11} + \frac{94078556297280}{172683145129} a^{10} - \frac{163979623616467}{172683145129} a^{9} + \frac{819816103341787}{518049435387} a^{8} - \frac{389270343852869}{172683145129} a^{7} + \frac{1381321028556226}{518049435387} a^{6} - \frac{1432602369033107}{518049435387} a^{5} + \frac{430736652616561}{172683145129} a^{4} - \frac{911851529094113}{518049435387} a^{3} + \frac{443298255381580}{518049435387} a^{2} - \frac{43388449242568}{172683145129} a + \frac{18125933597297}{518049435387} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 750418.760712 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.24217.1, 10.0.142510530627.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$3.10.5.2$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}$
61Data not computed
397Data not computed