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Properties

Label 20.0.43664893293...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 27517559^{2}$
Root discriminant $21.48$
Ramified primes $3, 5, 27517559$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T656

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 12, -15, 44, -42, 113, -63, 147, -43, 144, -12, 45, 14, 21, 8, 2, 2, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 5*x^18 + 2*x^17 + 2*x^16 + 8*x^15 + 21*x^14 + 14*x^13 + 45*x^12 - 12*x^11 + 144*x^10 - 43*x^9 + 147*x^8 - 63*x^7 + 113*x^6 - 42*x^5 + 44*x^4 - 15*x^3 + 12*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 5*x^18 + 2*x^17 + 2*x^16 + 8*x^15 + 21*x^14 + 14*x^13 + 45*x^12 - 12*x^11 + 144*x^10 - 43*x^9 + 147*x^8 - 63*x^7 + 113*x^6 - 42*x^5 + 44*x^4 - 15*x^3 + 12*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 5 x^{18} + 2 x^{17} + 2 x^{16} + 8 x^{15} + 21 x^{14} + 14 x^{13} + 45 x^{12} - 12 x^{11} + 144 x^{10} - 43 x^{9} + 147 x^{8} - 63 x^{7} + 113 x^{6} - 42 x^{5} + 44 x^{4} - 15 x^{3} + 12 x^{2} - 3 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(436648932933622896181640625=3^{10}\cdot 5^{10}\cdot 27517559^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.48$
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:  $3, 5, 27517559$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{11845037414953} a^{19} - \frac{4251585145408}{11845037414953} a^{18} - \frac{3493871650566}{11845037414953} a^{17} - \frac{1571344720462}{11845037414953} a^{16} + \frac{689049859313}{11845037414953} a^{15} + \frac{2302141905055}{11845037414953} a^{14} + \frac{5205314653625}{11845037414953} a^{13} + \frac{5634782736059}{11845037414953} a^{12} - \frac{1206122581855}{11845037414953} a^{11} - \frac{1909759651601}{11845037414953} a^{10} - \frac{4837045546219}{11845037414953} a^{9} - \frac{1378455818952}{11845037414953} a^{8} - \frac{5860630068548}{11845037414953} a^{7} - \frac{2873583381792}{11845037414953} a^{6} + \frac{1563294869133}{11845037414953} a^{5} - \frac{1275686212303}{11845037414953} a^{4} + \frac{1009813017576}{11845037414953} a^{3} - \frac{2416826229708}{11845037414953} a^{2} + \frac{338667135601}{11845037414953} a + \frac{4040077202071}{11845037414953}$

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:  \( \frac{4445828551289}{11845037414953} a^{19} - \frac{9115288207092}{11845037414953} a^{18} + \frac{22278689477411}{11845037414953} a^{17} + \frac{8991715116542}{11845037414953} a^{16} + \frac{5699800065648}{11845037414953} a^{15} + \frac{36096970502482}{11845037414953} a^{14} + \frac{92383871656097}{11845037414953} a^{13} + \frac{54744863726066}{11845037414953} a^{12} + \frac{191532866542614}{11845037414953} a^{11} - \frac{59102636321240}{11845037414953} a^{10} + \frac{632101248853966}{11845037414953} a^{9} - \frac{201474228570437}{11845037414953} a^{8} + \frac{603863491281561}{11845037414953} a^{7} - \frac{239814036465975}{11845037414953} a^{6} + \frac{457413820316399}{11845037414953} a^{5} - \frac{146772847114444}{11845037414953} a^{4} + \frac{152314241362690}{11845037414953} a^{3} - \frac{21259049322422}{11845037414953} a^{2} + \frac{38476179395432}{11845037414953} a + \frac{2606582401394}{11845037414953} \) (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:  \( 218812.807689 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T656:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 70 conjugacy class representatives for t20n656 are not computed
Character table for t20n656 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 10.8.85992371875.1

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 24 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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
3Data not computed
5Data not computed
27517559Data not computed