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Properties

Label 20.12.3639926562...2416.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{40}\cdot 97^{2}\cdot 2657^{6}$
Root discriminant $67.31$
Ramified primes $2, 97, 2657$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-22172191, 15782492, 49234722, -13729296, -11267540, 2865480, -9709878, 4954512, 1455417, -1578764, 732390, -139296, -69687, 45076, -12182, 1068, 564, -176, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 30*x^18 - 176*x^17 + 564*x^16 + 1068*x^15 - 12182*x^14 + 45076*x^13 - 69687*x^12 - 139296*x^11 + 732390*x^10 - 1578764*x^9 + 1455417*x^8 + 4954512*x^7 - 9709878*x^6 + 2865480*x^5 - 11267540*x^4 - 13729296*x^3 + 49234722*x^2 + 15782492*x - 22172191)
 
gp: K = bnfinit(x^20 - 8*x^19 + 30*x^18 - 176*x^17 + 564*x^16 + 1068*x^15 - 12182*x^14 + 45076*x^13 - 69687*x^12 - 139296*x^11 + 732390*x^10 - 1578764*x^9 + 1455417*x^8 + 4954512*x^7 - 9709878*x^6 + 2865480*x^5 - 11267540*x^4 - 13729296*x^3 + 49234722*x^2 + 15782492*x - 22172191, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 30 x^{18} - 176 x^{17} + 564 x^{16} + 1068 x^{15} - 12182 x^{14} + 45076 x^{13} - 69687 x^{12} - 139296 x^{11} + 732390 x^{10} - 1578764 x^{9} + 1455417 x^{8} + 4954512 x^{7} - 9709878 x^{6} + 2865480 x^{5} - 11267540 x^{4} - 13729296 x^{3} + 49234722 x^{2} + 15782492 x - 22172191 \)

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:  \(3639926562026986415967008755774652416=2^{40}\cdot 97^{2}\cdot 2657^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.31$
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:  $2, 97, 2657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{17} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{12} + \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{7179142385395305791140292866337336936282750780041296730223780396941037395892} a^{19} - \frac{619935862368994144515023166774414226714302069347039592023653452695397462493}{7179142385395305791140292866337336936282750780041296730223780396941037395892} a^{18} - \frac{299225028224201038927555939080548584925140699908274400470856924487684752367}{1794785596348826447785073216584334234070687695010324182555945099235259348973} a^{17} - \frac{416290814799081469889744226264326215236847851852371145917921367636273241467}{7179142385395305791140292866337336936282750780041296730223780396941037395892} a^{16} - \frac{2756939887197051633303624629378718831124991692378169344061794666839238413285}{7179142385395305791140292866337336936282750780041296730223780396941037395892} a^{15} + \frac{459497413280660497557242696567793318514547237264094598196924471093872385393}{3589571192697652895570146433168668468141375390020648365111890198470518697946} a^{14} + \frac{767883806803208885332814315993482761752833752972103133226695828188305726551}{7179142385395305791140292866337336936282750780041296730223780396941037395892} a^{13} - \frac{2434094690403570089141944526012460043326112095330777908753182356609125573143}{7179142385395305791140292866337336936282750780041296730223780396941037395892} a^{12} + \frac{2503108346773104873043461283374863725716541661208585037393820005284764596203}{7179142385395305791140292866337336936282750780041296730223780396941037395892} a^{11} - \frac{467065789737328792801549219398263523857997668257288865113711163978492291590}{1794785596348826447785073216584334234070687695010324182555945099235259348973} a^{10} + \frac{187298078360735653390376531983327203602596855331794427676513994290388007141}{7179142385395305791140292866337336936282750780041296730223780396941037395892} a^{9} - \frac{974511467865166286736635679029925412304502874647196830271968166427520825327}{7179142385395305791140292866337336936282750780041296730223780396941037395892} a^{8} - \frac{162971988165046698354867414650932423312668891226163437287405792354149319559}{7179142385395305791140292866337336936282750780041296730223780396941037395892} a^{7} - \frac{1753663151410266286880003190030135386206742139800942376007908679531742407589}{3589571192697652895570146433168668468141375390020648365111890198470518697946} a^{6} - \frac{792915457163888499536524062867761985233569080870321406662487058192058253059}{7179142385395305791140292866337336936282750780041296730223780396941037395892} a^{5} + \frac{312888630727210686836005553520484423767163934855397997789236959366765926639}{7179142385395305791140292866337336936282750780041296730223780396941037395892} a^{4} + \frac{464659041329855456692479859952939502639154751961216521137576692963316440451}{3589571192697652895570146433168668468141375390020648365111890198470518697946} a^{3} - \frac{2769647797376402905930178823932458492194605915534207894968192828438459478401}{7179142385395305791140292866337336936282750780041296730223780396941037395892} a^{2} - \frac{3580213602051610938724548986460789962542766586317291031726906866054076931779}{7179142385395305791140292866337336936282750780041296730223780396941037395892} a + \frac{844708448545155159962530278116295170351688078137208981805909933309356847779}{3589571192697652895570146433168668468141375390020648365111890198470518697946}$

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:  \( 226331742663 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.6.925322313728.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.12.24.336$x^{12} + 4 x^{11} - 2 x^{10} + 4 x^{6} + 4 x^{5} + 4 x^{4} - 2 x^{2} + 4 x - 2$$12$$1$$24$$C_2 \times S_4$$[4/3, 4/3, 3]_{3}^{2}$
97Data not computed
2657Data not computed