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Properties

Label 20.16.3142791914...8329.1
Degree $20$
Signature $[16, 2]$
Discriminant $3^{6}\cdot 401^{11}$
Root discriminant $37.57$
Ramified primes $3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times C_2^4:C_5).C_2$ (as 20T84)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 81, 710, -1579, -7292, 11099, 21932, -29369, -25374, 35908, 10712, -20884, -459, 5786, -505, -830, 73, 85, -7, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 7*x^18 + 85*x^17 + 73*x^16 - 830*x^15 - 505*x^14 + 5786*x^13 - 459*x^12 - 20884*x^11 + 10712*x^10 + 35908*x^9 - 25374*x^8 - 29369*x^7 + 21932*x^6 + 11099*x^5 - 7292*x^4 - 1579*x^3 + 710*x^2 + 81*x + 1)
 
gp: K = bnfinit(x^20 - 6*x^19 - 7*x^18 + 85*x^17 + 73*x^16 - 830*x^15 - 505*x^14 + 5786*x^13 - 459*x^12 - 20884*x^11 + 10712*x^10 + 35908*x^9 - 25374*x^8 - 29369*x^7 + 21932*x^6 + 11099*x^5 - 7292*x^4 - 1579*x^3 + 710*x^2 + 81*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 7 x^{18} + 85 x^{17} + 73 x^{16} - 830 x^{15} - 505 x^{14} + 5786 x^{13} - 459 x^{12} - 20884 x^{11} + 10712 x^{10} + 35908 x^{9} - 25374 x^{8} - 29369 x^{7} + 21932 x^{6} + 11099 x^{5} - 7292 x^{4} - 1579 x^{3} + 710 x^{2} + 81 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(31427919143590467585816658408329=3^{6}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.57$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{6} a^{16} + \frac{1}{6} a^{15} + \frac{1}{6} a^{14} - \frac{1}{6} a^{13} + \frac{1}{3} a^{12} - \frac{1}{2} a^{11} - \frac{1}{3} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{6} a^{17} - \frac{1}{3} a^{14} - \frac{1}{2} a^{13} + \frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{18} + \frac{1}{6} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{2} a^{8} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{25387989477577193899846531518} a^{19} + \frac{495144002571444455231562821}{8462663159192397966615510506} a^{18} + \frac{439066456986587618911058498}{12693994738788596949923265759} a^{17} + \frac{1728765431474784832497614893}{25387989477577193899846531518} a^{16} - \frac{736283788809026980411136488}{4231331579596198983307755253} a^{15} + \frac{4333860429110331035241257369}{25387989477577193899846531518} a^{14} + \frac{5805915964180361831217067159}{25387989477577193899846531518} a^{13} - \frac{5516890016468108199581838509}{12693994738788596949923265759} a^{12} + \frac{109951459572983652239008276}{4231331579596198983307755253} a^{11} + \frac{5249791682464699789226576525}{12693994738788596949923265759} a^{10} - \frac{12594922116352022324582816209}{25387989477577193899846531518} a^{9} - \frac{11754228462885916609700400847}{25387989477577193899846531518} a^{8} + \frac{429045562553341262785473331}{12693994738788596949923265759} a^{7} - \frac{3512917056681941374666617472}{12693994738788596949923265759} a^{6} + \frac{3020917721148013692042485939}{25387989477577193899846531518} a^{5} + \frac{27038029894460774826513017}{61771263935710934062886938} a^{4} - \frac{94694020063251198811420616}{12693994738788596949923265759} a^{3} - \frac{3142994836415897965217447513}{8462663159192397966615510506} a^{2} + \frac{8014176227480344143823025957}{25387989477577193899846531518} a + \frac{5541781157824457506213426399}{12693994738788596949923265759}$

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:  $17$
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:  \( 966729252.301 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_2^4:C_5).C_2$ (as 20T84):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $(C_2\times C_2^4:C_5).C_2$
Character table for $(C_2\times C_2^4:C_5).C_2$

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed