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Properties

Label 20.4.11449692206...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 5^{14}\cdot 17^{15}$
Root discriminant $44.97$
Ramified primes $2, 5, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![78434, -209480, -1960, 341222, -189846, 88253, -135103, -126527, 209295, -104641, 87067, -52413, -3031, 10887, -713, -1113, 67, 61, 13, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 13*x^18 + 61*x^17 + 67*x^16 - 1113*x^15 - 713*x^14 + 10887*x^13 - 3031*x^12 - 52413*x^11 + 87067*x^10 - 104641*x^9 + 209295*x^8 - 126527*x^7 - 135103*x^6 + 88253*x^5 - 189846*x^4 + 341222*x^3 - 1960*x^2 - 209480*x + 78434)
 
gp: K = bnfinit(x^20 - 9*x^19 + 13*x^18 + 61*x^17 + 67*x^16 - 1113*x^15 - 713*x^14 + 10887*x^13 - 3031*x^12 - 52413*x^11 + 87067*x^10 - 104641*x^9 + 209295*x^8 - 126527*x^7 - 135103*x^6 + 88253*x^5 - 189846*x^4 + 341222*x^3 - 1960*x^2 - 209480*x + 78434, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 13 x^{18} + 61 x^{17} + 67 x^{16} - 1113 x^{15} - 713 x^{14} + 10887 x^{13} - 3031 x^{12} - 52413 x^{11} + 87067 x^{10} - 104641 x^{9} + 209295 x^{8} - 126527 x^{7} - 135103 x^{6} + 88253 x^{5} - 189846 x^{4} + 341222 x^{3} - 1960 x^{2} - 209480 x + 78434 \)

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:  \(1144969220603926317200000000000000=2^{16}\cdot 5^{14}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.97$
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:  $2, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{13} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{8} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{816} a^{18} - \frac{25}{816} a^{17} - \frac{7}{136} a^{16} + \frac{3}{34} a^{15} + \frac{73}{816} a^{14} + \frac{47}{816} a^{13} - \frac{47}{408} a^{11} + \frac{35}{272} a^{10} + \frac{101}{816} a^{9} - \frac{67}{204} a^{8} + \frac{23}{51} a^{7} - \frac{79}{272} a^{6} - \frac{29}{272} a^{5} + \frac{59}{204} a^{4} - \frac{29}{136} a^{3} - \frac{137}{408} a^{2} + \frac{1}{17} a + \frac{179}{408}$, $\frac{1}{16951336872829553268028113009918528664416} a^{19} - \frac{5011450648592525765124971575512344773}{8475668436414776634014056504959264332208} a^{18} + \frac{198362610827885095363359196839786800491}{16951336872829553268028113009918528664416} a^{17} - \frac{349071351900434587155911091214559109671}{2825222812138258878004685501653088110736} a^{16} - \frac{1226379638384832081249692788505312947307}{16951336872829553268028113009918528664416} a^{15} - \frac{739598753889977774180704433400589528519}{8475668436414776634014056504959264332208} a^{14} - \frac{1542467388075526140307327426463574107015}{16951336872829553268028113009918528664416} a^{13} - \frac{10459082775792690156388573986212440877}{197108568288715735674745500115331728656} a^{12} + \frac{267427387044227615321937856538221202359}{16951336872829553268028113009918528664416} a^{11} - \frac{376637445989390076990748279288190592079}{4237834218207388317007028252479632166104} a^{10} + \frac{2210428664858553834951276053824278046457}{5650445624276517756009371003306176221472} a^{9} + \frac{114361977906203447759948623593104411417}{353152851517282359750585687706636013842} a^{8} - \frac{8260412237914488580989324454011358372145}{16951336872829553268028113009918528664416} a^{7} + \frac{1218446370055197476466855021402581422003}{2825222812138258878004685501653088110736} a^{6} - \frac{6251720297959485023945690467496857611637}{16951336872829553268028113009918528664416} a^{5} + \frac{270530800505021620377542185030348964927}{8475668436414776634014056504959264332208} a^{4} + \frac{525959977464063888068731453377018601691}{4237834218207388317007028252479632166104} a^{3} - \frac{1076207764316539209348684916200232674007}{8475668436414776634014056504959264332208} a^{2} - \frac{219913757890732939476482674089253505377}{8475668436414776634014056504959264332208} a - \frac{1945832153189499056072832722980381418891}{8475668436414776634014056504959264332208}$

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:  \( 2400799092.9352427 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.122825.1, 5.1.578000.2, 10.2.5679428000000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
5Data not computed
17Data not computed