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Properties

Label 20.4.49641192728...0000.2
Degree $20$
Signature $[4, 8]$
Discriminant $2^{24}\cdot 5^{10}\cdot 13^{6}\cdot 29^{4}\cdot 31^{6}$
Root discriminant $60.92$
Ramified primes $2, 5, 13, 29, 31$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![371831, -497699, -118358, 738530, 139931, -348632, -204976, 195260, -8639, 25565, 11468, -15830, -1741, -106, 262, 502, -83, 3, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 6*x^18 + 3*x^17 - 83*x^16 + 502*x^15 + 262*x^14 - 106*x^13 - 1741*x^12 - 15830*x^11 + 11468*x^10 + 25565*x^9 - 8639*x^8 + 195260*x^7 - 204976*x^6 - 348632*x^5 + 139931*x^4 + 738530*x^3 - 118358*x^2 - 497699*x + 371831)
 
gp: K = bnfinit(x^20 - 4*x^19 - 6*x^18 + 3*x^17 - 83*x^16 + 502*x^15 + 262*x^14 - 106*x^13 - 1741*x^12 - 15830*x^11 + 11468*x^10 + 25565*x^9 - 8639*x^8 + 195260*x^7 - 204976*x^6 - 348632*x^5 + 139931*x^4 + 738530*x^3 - 118358*x^2 - 497699*x + 371831, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 6 x^{18} + 3 x^{17} - 83 x^{16} + 502 x^{15} + 262 x^{14} - 106 x^{13} - 1741 x^{12} - 15830 x^{11} + 11468 x^{10} + 25565 x^{9} - 8639 x^{8} + 195260 x^{7} - 204976 x^{6} - 348632 x^{5} + 139931 x^{4} + 738530 x^{3} - 118358 x^{2} - 497699 x + 371831 \)

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:  \(496411927280973952950108160000000000=2^{24}\cdot 5^{10}\cdot 13^{6}\cdot 29^{4}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.92$
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:  $2, 5, 13, 29, 31$
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}{4235326140345022220326385714726327915762254781642914836831909929} a^{19} + \frac{1267805130129179944039459741165816917666558558859798318732670959}{4235326140345022220326385714726327915762254781642914836831909929} a^{18} - \frac{293047476880308729034848018772552887783374462453046554280668806}{4235326140345022220326385714726327915762254781642914836831909929} a^{17} + \frac{1341171064825772194969041824837154895070031594119558827799088841}{4235326140345022220326385714726327915762254781642914836831909929} a^{16} + \frac{2115238322571522643957979696042865015164862482509520789463035532}{4235326140345022220326385714726327915762254781642914836831909929} a^{15} - \frac{1219520680365925397039648988582977683260908685035721380949411583}{4235326140345022220326385714726327915762254781642914836831909929} a^{14} - \frac{865383407519223282060828134558259407158146474412506948782331672}{4235326140345022220326385714726327915762254781642914836831909929} a^{13} - \frac{1123237536824954658895454782680472742571726280947606461051593445}{4235326140345022220326385714726327915762254781642914836831909929} a^{12} + \frac{2046848143490335403657483963482137355823076385485772626974089818}{4235326140345022220326385714726327915762254781642914836831909929} a^{11} + \frac{433588513619157592465147524992155881605017525817668413470905506}{4235326140345022220326385714726327915762254781642914836831909929} a^{10} + \frac{1515620037857485814306432693870561126690555183430866772396937583}{4235326140345022220326385714726327915762254781642914836831909929} a^{9} - \frac{2085679878908605565343955349548677790918995406742965412712338785}{4235326140345022220326385714726327915762254781642914836831909929} a^{8} - \frac{1806688830953051413767379055499762808980053629417267639642740557}{4235326140345022220326385714726327915762254781642914836831909929} a^{7} - \frac{121664220017079526703725214916239815949915264874837333793563657}{4235326140345022220326385714726327915762254781642914836831909929} a^{6} - \frac{982166904909211665222287751418995426336857776315507946459863933}{4235326140345022220326385714726327915762254781642914836831909929} a^{5} - \frac{1353505354647231333435358776883720996230215050907928188835957359}{4235326140345022220326385714726327915762254781642914836831909929} a^{4} + \frac{71155317571133140888364905874678195407198809026812471397569773}{4235326140345022220326385714726327915762254781642914836831909929} a^{3} - \frac{665953640365622537124918173767773743752071892920402500686851678}{4235326140345022220326385714726327915762254781642914836831909929} a^{2} + \frac{312035678665002002076403256767445095637946690248473121360272470}{4235326140345022220326385714726327915762254781642914836831909929} a - \frac{1419871474929738180879595781036992993396031142756008669960763032}{4235326140345022220326385714726327915762254781642914836831909929}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 2078876430.66 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.109268775200000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.10.0.1$x^{10} + x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
31Data not computed