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Properties

Label 20.12.1753321685...3125.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{10}\cdot 5^{11}\cdot 239^{10}$
Root discriminant $64.89$
Ramified primes $3, 5, 239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -99, -6201, -65625, -260864, -346819, 93723, 257416, -132368, -123986, 88202, 24039, -31742, 1761, 5814, -1468, -394, 207, -7, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 7*x^18 + 207*x^17 - 394*x^16 - 1468*x^15 + 5814*x^14 + 1761*x^13 - 31742*x^12 + 24039*x^11 + 88202*x^10 - 123986*x^9 - 132368*x^8 + 257416*x^7 + 93723*x^6 - 346819*x^5 - 260864*x^4 - 65625*x^3 - 6201*x^2 - 99*x + 9)
 
gp: K = bnfinit(x^20 - 8*x^19 - 7*x^18 + 207*x^17 - 394*x^16 - 1468*x^15 + 5814*x^14 + 1761*x^13 - 31742*x^12 + 24039*x^11 + 88202*x^10 - 123986*x^9 - 132368*x^8 + 257416*x^7 + 93723*x^6 - 346819*x^5 - 260864*x^4 - 65625*x^3 - 6201*x^2 - 99*x + 9, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} - 7 x^{18} + 207 x^{17} - 394 x^{16} - 1468 x^{15} + 5814 x^{14} + 1761 x^{13} - 31742 x^{12} + 24039 x^{11} + 88202 x^{10} - 123986 x^{9} - 132368 x^{8} + 257416 x^{7} + 93723 x^{6} - 346819 x^{5} - 260864 x^{4} - 65625 x^{3} - 6201 x^{2} - 99 x + 9 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1753321685810638349237472141064453125=3^{10}\cdot 5^{11}\cdot 239^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.89$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{15} a^{16} - \frac{2}{15} a^{14} - \frac{1}{5} a^{13} - \frac{7}{15} a^{12} - \frac{1}{15} a^{11} + \frac{1}{15} a^{10} + \frac{4}{15} a^{9} + \frac{2}{15} a^{8} - \frac{1}{3} a^{7} + \frac{7}{15} a^{6} + \frac{2}{5} a^{5} + \frac{2}{15} a^{4} - \frac{1}{15} a^{3} - \frac{7}{15} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{15} a^{17} - \frac{2}{15} a^{15} + \frac{2}{15} a^{14} - \frac{7}{15} a^{13} - \frac{2}{5} a^{12} + \frac{1}{15} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{3} a^{8} + \frac{7}{15} a^{7} - \frac{4}{15} a^{6} + \frac{7}{15} a^{5} - \frac{2}{5} a^{4} - \frac{2}{15} a^{3} - \frac{7}{15} a^{2} + \frac{2}{5} a$, $\frac{1}{15} a^{18} + \frac{2}{15} a^{15} - \frac{1}{15} a^{14} + \frac{1}{5} a^{13} + \frac{7}{15} a^{12} + \frac{7}{15} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{15} a^{8} + \frac{1}{15} a^{7} + \frac{1}{15} a^{6} + \frac{1}{15} a^{5} + \frac{7}{15} a^{4} + \frac{1}{15} a^{3} + \frac{2}{15} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{203405272628436225187946557800927404861955} a^{19} + \frac{929778390272322130093197100667771460479}{203405272628436225187946557800927404861955} a^{18} - \frac{254640175988609899759551669432032220097}{67801757542812075062648852600309134953985} a^{17} + \frac{1891171514126043627109956173927354133989}{67801757542812075062648852600309134953985} a^{16} + \frac{1420238639276505307195394680490083285783}{67801757542812075062648852600309134953985} a^{15} - \frac{13130250305817843515602520956977422911958}{203405272628436225187946557800927404861955} a^{14} - \frac{30756785607649863301478333539518952104959}{203405272628436225187946557800927404861955} a^{13} + \frac{4720398388922956585675251304618808134247}{67801757542812075062648852600309134953985} a^{12} + \frac{32010455845800426069144564196674137449156}{203405272628436225187946557800927404861955} a^{11} + \frac{2317001539001108711182532698330705574443}{13560351508562415012529770520061826990797} a^{10} + \frac{21079627894618636379729166057227641604827}{67801757542812075062648852600309134953985} a^{9} + \frac{6600087247609181014697450334136683047460}{13560351508562415012529770520061826990797} a^{8} - \frac{88287013108614185128665858814362605421107}{203405272628436225187946557800927404861955} a^{7} + \frac{13712100245637253395266541954746946659673}{67801757542812075062648852600309134953985} a^{6} - \frac{477677885315442248878667039312190025486}{203405272628436225187946557800927404861955} a^{5} - \frac{1218834550211924693544894359215321280865}{13560351508562415012529770520061826990797} a^{4} - \frac{26137126788079603843351008230688805143517}{203405272628436225187946557800927404861955} a^{3} + \frac{48738271706404595032287669643889297945381}{203405272628436225187946557800927404861955} a^{2} + \frac{5695668451862449238270087553059788664918}{13560351508562415012529770520061826990797} a + \frac{10393649871602568735814513068304448152391}{67801757542812075062648852600309134953985}$

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:  $15$
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:  \( 134860480257 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 $20$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
239Data not computed