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Properties

Label 20.0.85672046165...0625.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 41^{5}\cdot 27517559^{2}$
Root discriminant $31.38$
Ramified primes $5, 41, 27517559$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group 20T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, -2048, 5888, -4864, 11072, -4384, 11616, -1232, 7876, 330, 3309, -11, -55, -518, -146, -28, 17, 2, 3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 3*x^18 + 2*x^17 + 17*x^16 - 28*x^15 - 146*x^14 - 518*x^13 - 55*x^12 - 11*x^11 + 3309*x^10 + 330*x^9 + 7876*x^8 - 1232*x^7 + 11616*x^6 - 4384*x^5 + 11072*x^4 - 4864*x^3 + 5888*x^2 - 2048*x + 1024)
 
gp: K = bnfinit(x^20 - x^19 + 3*x^18 + 2*x^17 + 17*x^16 - 28*x^15 - 146*x^14 - 518*x^13 - 55*x^12 - 11*x^11 + 3309*x^10 + 330*x^9 + 7876*x^8 - 1232*x^7 + 11616*x^6 - 4384*x^5 + 11072*x^4 - 4864*x^3 + 5888*x^2 - 2048*x + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 3 x^{18} + 2 x^{17} + 17 x^{16} - 28 x^{15} - 146 x^{14} - 518 x^{13} - 55 x^{12} - 11 x^{11} + 3309 x^{10} + 330 x^{9} + 7876 x^{8} - 1232 x^{7} + 11616 x^{6} - 4384 x^{5} + 11072 x^{4} - 4864 x^{3} + 5888 x^{2} - 2048 x + 1024 \)

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:  \(856720461657154802252744140625=5^{10}\cdot 41^{5}\cdot 27517559^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.38$
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:  $5, 41, 27517559$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{3}{8} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{3}{8} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{8} a^{11} - \frac{3}{16} a^{10} + \frac{1}{4} a^{9} - \frac{3}{8} a^{8} + \frac{1}{8} a^{7} + \frac{1}{16} a^{6} - \frac{3}{16} a^{5} - \frac{7}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{14} - \frac{1}{32} a^{13} - \frac{1}{16} a^{12} - \frac{3}{32} a^{11} - \frac{3}{8} a^{10} + \frac{5}{16} a^{9} + \frac{1}{16} a^{8} + \frac{1}{32} a^{7} - \frac{3}{32} a^{6} - \frac{7}{32} a^{5} + \frac{7}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{16} - \frac{1}{64} a^{15} - \frac{1}{64} a^{14} - \frac{1}{32} a^{13} - \frac{3}{64} a^{12} - \frac{3}{16} a^{11} + \frac{5}{32} a^{10} - \frac{15}{32} a^{9} - \frac{31}{64} a^{8} - \frac{3}{64} a^{7} + \frac{25}{64} a^{6} + \frac{7}{32} a^{5} + \frac{3}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{128} a^{17} - \frac{1}{128} a^{16} - \frac{1}{128} a^{15} - \frac{1}{64} a^{14} - \frac{3}{128} a^{13} - \frac{3}{32} a^{12} + \frac{5}{64} a^{11} - \frac{15}{64} a^{10} - \frac{31}{128} a^{9} - \frac{3}{128} a^{8} - \frac{39}{128} a^{7} - \frac{25}{64} a^{6} + \frac{3}{16} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{256} a^{18} - \frac{1}{256} a^{17} - \frac{1}{256} a^{16} - \frac{1}{128} a^{15} - \frac{3}{256} a^{14} - \frac{3}{64} a^{13} + \frac{5}{128} a^{12} - \frac{15}{128} a^{11} - \frac{31}{256} a^{10} - \frac{3}{256} a^{9} - \frac{39}{256} a^{8} + \frac{39}{128} a^{7} - \frac{13}{32} a^{6} - \frac{1}{16} a^{5} + \frac{1}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10458036823855541835309743053312} a^{19} + \frac{7522386402913710827770984839}{10458036823855541835309743053312} a^{18} + \frac{2420081481809693174887741015}{10458036823855541835309743053312} a^{17} + \frac{23565448617715584333078695563}{5229018411927770917654871526656} a^{16} - \frac{64567349614688284149262254243}{10458036823855541835309743053312} a^{15} + \frac{27302364025041440448132754827}{2614509205963885458827435763328} a^{14} + \frac{418167022263749502555204189}{15516375109577955245266681088} a^{13} - \frac{579513204224601517444806067047}{5229018411927770917654871526656} a^{12} - \frac{1913922268535284816395527117567}{10458036823855541835309743053312} a^{11} + \frac{577525642128104559845428067557}{10458036823855541835309743053312} a^{10} + \frac{5136725245105945914232835151169}{10458036823855541835309743053312} a^{9} - \frac{1655917962768380482319131085973}{5229018411927770917654871526656} a^{8} + \frac{573200398987968332010889216959}{1307254602981942729413717881664} a^{7} - \frac{132790530512355910947768127129}{326813650745485682353429470416} a^{6} - \frac{51245613625996307410515335761}{163406825372742841176714735208} a^{5} - \frac{5470415275974849540985095485}{81703412686371420588357367604} a^{4} + \frac{44205542172299765063590570059}{163406825372742841176714735208} a^{3} + \frac{8095847326762205460870866441}{81703412686371420588357367604} a^{2} + \frac{9673387962266240531857917165}{40851706343185710294178683802} a + \frac{2507103230762355567291343852}{20425853171592855147089341901}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{7}$, which has order $7$ (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:  \( -1 \) (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:  \( 446422.774797 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 115200
The 119 conjugacy class representatives for t20n781 are not computed
Character table for t20n781 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.1025.1, 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
27517559Data not computed