↑

Properties

Label 20.20.1678055009...3488.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 11^{16}\cdot 23^{5}\cdot 419^{6}$
Root discriminant $182.49$
Ramified primes $2, 11, 23, 419$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T432

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![375007194660407, 0, -438902785988869, 0, 150667328531683, 0, -21079412126729, 0, 1451946981565, 0, -55069019284, 0, 1223654732, 0, -16333702, 0, 128883, 0, -554, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 554*x^18 + 128883*x^16 - 16333702*x^14 + 1223654732*x^12 - 55069019284*x^10 + 1451946981565*x^8 - 21079412126729*x^6 + 150667328531683*x^4 - 438902785988869*x^2 + 375007194660407)
 
gp: K = bnfinit(x^20 - 554*x^18 + 128883*x^16 - 16333702*x^14 + 1223654732*x^12 - 55069019284*x^10 + 1451946981565*x^8 - 21079412126729*x^6 + 150667328531683*x^4 - 438902785988869*x^2 + 375007194660407, 1)
 

Normalized defining polynomial

\( x^{20} - 554 x^{18} + 128883 x^{16} - 16333702 x^{14} + 1223654732 x^{12} - 55069019284 x^{10} + 1451946981565 x^{8} - 21079412126729 x^{6} + 150667328531683 x^{4} - 438902785988869 x^{2} + 375007194660407 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1678055009725811496947195505354733123810623488=2^{20}\cdot 11^{16}\cdot 23^{5}\cdot 419^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.49$
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:  $2, 11, 23, 419$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{86733} a^{14} + \frac{703}{86733} a^{12} - \frac{3138}{9637} a^{10} + \frac{4388}{28911} a^{8} - \frac{4187}{28911} a^{6} + \frac{13703}{86733} a^{4} - \frac{745}{3771} a^{2} + \frac{4}{9}$, $\frac{1}{86733} a^{15} + \frac{703}{86733} a^{13} - \frac{3138}{9637} a^{11} + \frac{4388}{28911} a^{9} - \frac{4187}{28911} a^{7} + \frac{13703}{86733} a^{5} - \frac{745}{3771} a^{3} + \frac{4}{9} a$, $\frac{1}{109023381} a^{16} - \frac{5}{4037903} a^{14} + \frac{423440}{109023381} a^{12} + \frac{9454514}{36341127} a^{10} - \frac{7526494}{36341127} a^{8} - \frac{53816065}{109023381} a^{6} - \frac{8551327}{109023381} a^{4} + \frac{245}{1257} a^{2} - \frac{8}{27}$, $\frac{1}{109023381} a^{17} - \frac{5}{4037903} a^{15} + \frac{423440}{109023381} a^{13} + \frac{9454514}{36341127} a^{11} - \frac{7526494}{36341127} a^{9} - \frac{53816065}{109023381} a^{7} - \frac{8551327}{109023381} a^{5} + \frac{245}{1257} a^{3} - \frac{8}{27} a$, $\frac{1}{4073220956251431617914640041824909237342015218339799} a^{18} + \frac{14528115169299073411789021845378845510173613}{4073220956251431617914640041824909237342015218339799} a^{16} - \frac{187891915939619264190205075964702916634507927}{60794342630618382356934925997386705034955451019997} a^{14} + \frac{590221246803504327355485132889278363414555828316723}{4073220956251431617914640041824909237342015218339799} a^{12} + \frac{61329749796941873466201311112366427642783860157112}{452580106250159068657182226869434359704668357593311} a^{10} + \frac{1377010814461343468574852191710389630272069102853494}{4073220956251431617914640041824909237342015218339799} a^{8} - \frac{455682838330917159545854059340999066274043104467246}{1357740318750477205971546680608303079114005072779933} a^{6} - \frac{1151266736141702318835289022972686318282229837}{422664828914748533559680402804286524576322010827} a^{4} + \frac{403905632810784186491617542983505615862034740}{1008746608388421321144821963733380726912463033} a^{2} - \frac{12294143229357826064359037268849613922974}{104674339357520112186865410784827303819909}$, $\frac{1}{4073220956251431617914640041824909237342015218339799} a^{19} + \frac{14528115169299073411789021845378845510173613}{4073220956251431617914640041824909237342015218339799} a^{17} - \frac{187891915939619264190205075964702916634507927}{60794342630618382356934925997386705034955451019997} a^{15} + \frac{590221246803504327355485132889278363414555828316723}{4073220956251431617914640041824909237342015218339799} a^{13} + \frac{61329749796941873466201311112366427642783860157112}{452580106250159068657182226869434359704668357593311} a^{11} + \frac{1377010814461343468574852191710389630272069102853494}{4073220956251431617914640041824909237342015218339799} a^{9} - \frac{455682838330917159545854059340999066274043104467246}{1357740318750477205971546680608303079114005072779933} a^{7} - \frac{1151266736141702318835289022972686318282229837}{422664828914748533559680402804286524576322010827} a^{5} + \frac{403905632810784186491617542983505615862034740}{1008746608388421321144821963733380726912463033} a^{3} - \frac{12294143229357826064359037268849613922974}{104674339357520112186865410784827303819909} a$

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:  $19$
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:  \( 19129848549700000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T432:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n432 are not computed
Character table for t20n432 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.2065776536197.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
2Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
419Data not computed