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Properties

Label 20.0.11194501700...1613.1
Degree $20$
Signature $[0, 10]$
Discriminant $7^{15}\cdot 11^{9}$
Root discriminant $12.66$
Ramified primes $7, 11$
Class number $1$
Class group Trivial
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37, -152, 258, -264, 222, -133, 12, 13, 17, 9, -38, 39, -29, -7, 46, -55, 43, -26, 13, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 13*x^18 - 26*x^17 + 43*x^16 - 55*x^15 + 46*x^14 - 7*x^13 - 29*x^12 + 39*x^11 - 38*x^10 + 9*x^9 + 17*x^8 + 13*x^7 + 12*x^6 - 133*x^5 + 222*x^4 - 264*x^3 + 258*x^2 - 152*x + 37)
 
gp: K = bnfinit(x^20 - 5*x^19 + 13*x^18 - 26*x^17 + 43*x^16 - 55*x^15 + 46*x^14 - 7*x^13 - 29*x^12 + 39*x^11 - 38*x^10 + 9*x^9 + 17*x^8 + 13*x^7 + 12*x^6 - 133*x^5 + 222*x^4 - 264*x^3 + 258*x^2 - 152*x + 37, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 13 x^{18} - 26 x^{17} + 43 x^{16} - 55 x^{15} + 46 x^{14} - 7 x^{13} - 29 x^{12} + 39 x^{11} - 38 x^{10} + 9 x^{9} + 17 x^{8} + 13 x^{7} + 12 x^{6} - 133 x^{5} + 222 x^{4} - 264 x^{3} + 258 x^{2} - 152 x + 37 \)

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:  \(11194501700250570391613=7^{15}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.66$
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:  $7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3379128342642024683} a^{19} - \frac{468946706865297903}{3379128342642024683} a^{18} - \frac{310231883188928737}{3379128342642024683} a^{17} - \frac{1190739733927385324}{3379128342642024683} a^{16} + \frac{1612635493344139968}{3379128342642024683} a^{15} - \frac{1599085054782823371}{3379128342642024683} a^{14} + \frac{1520037981764935983}{3379128342642024683} a^{13} - \frac{136642957347389353}{3379128342642024683} a^{12} + \frac{1117795633481043942}{3379128342642024683} a^{11} - \frac{1110322494479947558}{3379128342642024683} a^{10} - \frac{729213575623973233}{3379128342642024683} a^{9} + \frac{166757011572527558}{3379128342642024683} a^{8} - \frac{889352187779792049}{3379128342642024683} a^{7} - \frac{1636643948931667517}{3379128342642024683} a^{6} + \frac{311347860973452161}{3379128342642024683} a^{5} + \frac{299064738217427580}{3379128342642024683} a^{4} + \frac{1102777047629631916}{3379128342642024683} a^{3} + \frac{1326562002538863165}{3379128342642024683} a^{2} - \frac{698094419286338676}{3379128342642024683} a - \frac{890022352008737346}{3379128342642024683}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 412.2805664510847 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.3773.1, 10.0.246071287.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ $20$ $20$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ $20$

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
7Data not computed
$11$11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$