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Properties

Label 20.4.36006738474...5921.3
Degree $20$
Signature $[4, 8]$
Discriminant $17^{4}\cdot 401^{11}$
Root discriminant $47.62$
Ramified primes $17, 401$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T350

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-524809, 1264948, -1901083, -2070773, 785364, -1062570, -263315, 173129, -235873, 82259, -19841, -184, 9662, -4167, 3445, -1008, 527, -105, 38, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 38*x^18 - 105*x^17 + 527*x^16 - 1008*x^15 + 3445*x^14 - 4167*x^13 + 9662*x^12 - 184*x^11 - 19841*x^10 + 82259*x^9 - 235873*x^8 + 173129*x^7 - 263315*x^6 - 1062570*x^5 + 785364*x^4 - 2070773*x^3 - 1901083*x^2 + 1264948*x - 524809)
 
gp: K = bnfinit(x^20 - 4*x^19 + 38*x^18 - 105*x^17 + 527*x^16 - 1008*x^15 + 3445*x^14 - 4167*x^13 + 9662*x^12 - 184*x^11 - 19841*x^10 + 82259*x^9 - 235873*x^8 + 173129*x^7 - 263315*x^6 - 1062570*x^5 + 785364*x^4 - 2070773*x^3 - 1901083*x^2 + 1264948*x - 524809, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 38 x^{18} - 105 x^{17} + 527 x^{16} - 1008 x^{15} + 3445 x^{14} - 4167 x^{13} + 9662 x^{12} - 184 x^{11} - 19841 x^{10} + 82259 x^{9} - 235873 x^{8} + 173129 x^{7} - 263315 x^{6} - 1062570 x^{5} + 785364 x^{4} - 2070773 x^{3} - 1901083 x^{2} + 1264948 x - 524809 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3600673847451055477688605112375921=17^{4}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.62$
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:  $17, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{13} + \frac{2}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{2}{9} a^{7} - \frac{2}{9} a^{5} + \frac{2}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{9} a^{2} + \frac{2}{9} a - \frac{1}{9}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{14} + \frac{2}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{2}{9} a^{8} - \frac{2}{9} a^{6} + \frac{2}{9} a^{5} - \frac{1}{3} a^{4} - \frac{1}{9} a^{3} + \frac{2}{9} a^{2} - \frac{1}{9} a$, $\frac{1}{27} a^{18} + \frac{1}{27} a^{17} - \frac{1}{27} a^{16} - \frac{1}{27} a^{15} - \frac{1}{27} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{3} a^{11} - \frac{1}{27} a^{10} + \frac{4}{9} a^{8} + \frac{5}{27} a^{7} - \frac{1}{3} a^{6} + \frac{10}{27} a^{5} - \frac{2}{9} a^{4} + \frac{13}{27} a^{3} - \frac{7}{27} a^{2} - \frac{4}{9} a + \frac{1}{27}$, $\frac{1}{57929338163475793678882176491052519624840912995844789} a^{19} + \frac{245927860533309157444879549927844061746589345893659}{19309779387825264559627392163684173208280304331948263} a^{18} - \frac{2748139991504259625715058385252743142763376366739136}{57929338163475793678882176491052519624840912995844789} a^{17} - \frac{958500839343485199690854464919403789857776948177057}{19309779387825264559627392163684173208280304331948263} a^{16} - \frac{851235758431317459737685959984142376803828314631061}{19309779387825264559627392163684173208280304331948263} a^{15} + \frac{8237766426538484495876560698319108740994116025549375}{57929338163475793678882176491052519624840912995844789} a^{14} - \frac{822822543768736360454583259083511754175092565853069}{6436593129275088186542464054561391069426768110649421} a^{13} - \frac{1168020167302146298888106327625082633735134303541802}{19309779387825264559627392163684173208280304331948263} a^{12} + \frac{10583634426776162017373751973030819028247577935292046}{57929338163475793678882176491052519624840912995844789} a^{11} - \frac{13378343121790454368188762944251417579606117842306152}{57929338163475793678882176491052519624840912995844789} a^{10} + \frac{546668064427721754710777436768378621562373353930598}{6436593129275088186542464054561391069426768110649421} a^{9} - \frac{23588943443323874522694847009509241502449082979032050}{57929338163475793678882176491052519624840912995844789} a^{8} + \frac{22291460622257768205022186961092978784065145447641831}{57929338163475793678882176491052519624840912995844789} a^{7} + \frac{5661997255628972067203754408908642964256931816668570}{57929338163475793678882176491052519624840912995844789} a^{6} - \frac{20004998472349102027135180023481825997735893227756784}{57929338163475793678882176491052519624840912995844789} a^{5} - \frac{8054573284730005144021889418607208428824336061354112}{57929338163475793678882176491052519624840912995844789} a^{4} + \frac{22522148856322576388780802419951884333364080773687937}{57929338163475793678882176491052519624840912995844789} a^{3} + \frac{4640125526414632488150677293489835911659294978470233}{57929338163475793678882176491052519624840912995844789} a^{2} + \frac{22230849313387337130073463668542425112560301221517682}{57929338163475793678882176491052519624840912995844789} a + \frac{160128263621730704365109654355192869811792645336205}{697943833294889080468459957723524332829408590311383}$

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:  $11$
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:  \( 333861139.455 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T350:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n350 are not computed
Character table for t20n350 is not computed

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
401Data not computed