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Properties

Label 20.6.39582054809...4651.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,11^{17}\cdot 23^{8}$
Root discriminant $26.91$
Ramified primes $11, 23$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T310

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -118, 442, 1376, -1189, -2694, 2739, 1947, -3180, -290, 2177, -639, -989, 993, -389, 73, 0, -19, 18, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 18*x^18 - 19*x^17 + 73*x^15 - 389*x^14 + 993*x^13 - 989*x^12 - 639*x^11 + 2177*x^10 - 290*x^9 - 3180*x^8 + 1947*x^7 + 2739*x^6 - 2694*x^5 - 1189*x^4 + 1376*x^3 + 442*x^2 - 118*x + 1)
 
gp: K = bnfinit(x^20 - 7*x^19 + 18*x^18 - 19*x^17 + 73*x^15 - 389*x^14 + 993*x^13 - 989*x^12 - 639*x^11 + 2177*x^10 - 290*x^9 - 3180*x^8 + 1947*x^7 + 2739*x^6 - 2694*x^5 - 1189*x^4 + 1376*x^3 + 442*x^2 - 118*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 18 x^{18} - 19 x^{17} + 73 x^{15} - 389 x^{14} + 993 x^{13} - 989 x^{12} - 639 x^{11} + 2177 x^{10} - 290 x^{9} - 3180 x^{8} + 1947 x^{7} + 2739 x^{6} - 2694 x^{5} - 1189 x^{4} + 1376 x^{3} + 442 x^{2} - 118 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-39582054809133382019675984651=-\,11^{17}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.91$
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:  $11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{29759591430394807263952165187} a^{19} + \frac{3279802667582022463851239790}{29759591430394807263952165187} a^{18} - \frac{14052147852942873397357596961}{29759591430394807263952165187} a^{17} - \frac{9249354367272424340981268204}{29759591430394807263952165187} a^{16} + \frac{8001325490455282310680525832}{29759591430394807263952165187} a^{15} - \frac{8856752983740594879039540526}{29759591430394807263952165187} a^{14} + \frac{14861397011816170502625399987}{29759591430394807263952165187} a^{13} - \frac{7823218283765701684213027571}{29759591430394807263952165187} a^{12} + \frac{10293297536891084188278242938}{29759591430394807263952165187} a^{11} - \frac{11798989378329233090327725940}{29759591430394807263952165187} a^{10} - \frac{2425520069531818390996073925}{29759591430394807263952165187} a^{9} + \frac{4690734429805655556603167890}{29759591430394807263952165187} a^{8} + \frac{4552446985280652431510876016}{29759591430394807263952165187} a^{7} + \frac{13837201604051343148455248575}{29759591430394807263952165187} a^{6} + \frac{10672844488156517162253064275}{29759591430394807263952165187} a^{5} + \frac{14531038749343127591971371902}{29759591430394807263952165187} a^{4} - \frac{3566786941532674496993092774}{29759591430394807263952165187} a^{3} - \frac{1952619158334285529367708991}{29759591430394807263952165187} a^{2} + \frac{5221170109595141110820433182}{29759591430394807263952165187} a + \frac{10311337767389471444562475887}{29759591430394807263952165187}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $12$
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:  \( 2139066.25714 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T310:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 44 conjugacy class representatives for t20n310
Character table for t20n310 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.113395848049.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{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}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$