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Properties

Label 20.6.16405056404...2992.2
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{20}\cdot 11^{16}\cdot 23^{7}$
Root discriminant $40.81$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T749

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![150457, -519028, 426643, 463204, -1132323, 782588, 74459, -566768, 491401, -207504, 21352, 28030, -19377, 7066, -1951, 512, -80, -32, 27, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 27*x^18 - 32*x^17 - 80*x^16 + 512*x^15 - 1951*x^14 + 7066*x^13 - 19377*x^12 + 28030*x^11 + 21352*x^10 - 207504*x^9 + 491401*x^8 - 566768*x^7 + 74459*x^6 + 782588*x^5 - 1132323*x^4 + 463204*x^3 + 426643*x^2 - 519028*x + 150457)
 
gp: K = bnfinit(x^20 - 8*x^19 + 27*x^18 - 32*x^17 - 80*x^16 + 512*x^15 - 1951*x^14 + 7066*x^13 - 19377*x^12 + 28030*x^11 + 21352*x^10 - 207504*x^9 + 491401*x^8 - 566768*x^7 + 74459*x^6 + 782588*x^5 - 1132323*x^4 + 463204*x^3 + 426643*x^2 - 519028*x + 150457, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 27 x^{18} - 32 x^{17} - 80 x^{16} + 512 x^{15} - 1951 x^{14} + 7066 x^{13} - 19377 x^{12} + 28030 x^{11} + 21352 x^{10} - 207504 x^{9} + 491401 x^{8} - 566768 x^{7} + 74459 x^{6} + 782588 x^{5} - 1132323 x^{4} + 463204 x^{3} + 426643 x^{2} - 519028 x + 150457 \)

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:  \(-164050564045619941441358756052992=-\,2^{20}\cdot 11^{16}\cdot 23^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.81$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} - \frac{3}{11} a^{14} + \frac{2}{11} a^{13} - \frac{4}{11} a^{12} - \frac{1}{11} a^{11} - \frac{3}{11} a^{9} + \frac{1}{11} a^{8} - \frac{3}{11} a^{7} + \frac{1}{11} a^{6} - \frac{5}{11} a^{5} + \frac{4}{11} a^{4} - \frac{5}{11} a^{3} + \frac{5}{11} a^{2} - \frac{2}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{16} + \frac{4}{11} a^{14} + \frac{2}{11} a^{13} - \frac{2}{11} a^{12} - \frac{3}{11} a^{11} - \frac{3}{11} a^{10} + \frac{3}{11} a^{9} + \frac{3}{11} a^{7} - \frac{2}{11} a^{6} - \frac{4}{11} a^{4} + \frac{1}{11} a^{3} + \frac{2}{11} a^{2} - \frac{5}{11} a + \frac{3}{11}$, $\frac{1}{11} a^{17} + \frac{3}{11} a^{14} + \frac{1}{11} a^{13} + \frac{2}{11} a^{12} + \frac{1}{11} a^{11} + \frac{3}{11} a^{10} + \frac{1}{11} a^{9} - \frac{1}{11} a^{8} - \frac{1}{11} a^{7} - \frac{4}{11} a^{6} + \frac{5}{11} a^{5} - \frac{4}{11} a^{4} - \frac{3}{11} a^{2} - \frac{4}{11}$, $\frac{1}{40051} a^{18} - \frac{622}{40051} a^{17} + \frac{168}{40051} a^{16} - \frac{900}{40051} a^{15} + \frac{9370}{40051} a^{14} - \frac{316}{3641} a^{13} + \frac{7885}{40051} a^{12} + \frac{1203}{3641} a^{11} + \frac{15407}{40051} a^{10} - \frac{11402}{40051} a^{9} - \frac{2900}{40051} a^{8} + \frac{36}{121} a^{7} - \frac{5}{11} a^{6} + \frac{235}{40051} a^{5} + \frac{10271}{40051} a^{4} - \frac{15978}{40051} a^{3} + \frac{14990}{40051} a^{2} - \frac{16583}{40051} a + \frac{9459}{40051}$, $\frac{1}{374387184505309063414590293239029712093333} a^{19} - \frac{23542105368308263574180206065906812}{374387184505309063414590293239029712093333} a^{18} + \frac{3005692186421383200884599114056641262052}{374387184505309063414590293239029712093333} a^{17} - \frac{12155660174601061174716783566914107915285}{374387184505309063414590293239029712093333} a^{16} - \frac{4051777466727031860944816711047805675191}{374387184505309063414590293239029712093333} a^{15} + \frac{12752771363749919945352987133481071186073}{374387184505309063414590293239029712093333} a^{14} + \frac{149901237522287284032421107484811453463713}{374387184505309063414590293239029712093333} a^{13} + \frac{121186971015744946276805871471623822720119}{374387184505309063414590293239029712093333} a^{12} - \frac{40866437438051942530228514396358500440489}{374387184505309063414590293239029712093333} a^{11} - \frac{68722104361337906061411702610468707449644}{374387184505309063414590293239029712093333} a^{10} - \frac{7649071811866753245580480611280305799057}{34035198591391733037690026658093610190303} a^{9} + \frac{116295413431863241745170542235485201473911}{374387184505309063414590293239029712093333} a^{8} + \frac{227214297171047168410518482484558159212}{1131079107266794753518399677459304266143} a^{7} - \frac{15422590504159150515992771283808447578953}{374387184505309063414590293239029712093333} a^{6} + \frac{25507227566286892449067476163827837323776}{374387184505309063414590293239029712093333} a^{5} + \frac{175659461603037815721829991455591830738868}{374387184505309063414590293239029712093333} a^{4} - \frac{3706033323998650599785769172051775427928}{16277703674143872322373491010392596177971} a^{3} + \frac{17233225644850869186953457346384933214217}{374387184505309063414590293239029712093333} a^{2} - \frac{59318138461315411796725648848093648597890}{374387184505309063414590293239029712093333} a - \frac{53717039283642984480316570201332540574504}{374387184505309063414590293239029712093333}$

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:  $12$
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:  \( 236855992.505 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T749:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n749 are not computed
Character table for t20n749 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.116117348402176.2

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
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}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$