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Properties

Label 20.0.55374574082...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{5}\cdot 61^{10}\cdot 397^{4}$
Root discriminant $38.65$
Ramified primes $5, 61, 397$
Class number $16$ (GRH)
Class group $[16]$ (GRH)
Galois group 20T174

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19321, -46426, 51631, -18172, -21536, 30774, 7631, -31548, 19103, 4885, -6369, -167, 2989, -1487, 443, -8, 3, -35, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 18*x^18 - 35*x^17 + 3*x^16 - 8*x^15 + 443*x^14 - 1487*x^13 + 2989*x^12 - 167*x^11 - 6369*x^10 + 4885*x^9 + 19103*x^8 - 31548*x^7 + 7631*x^6 + 30774*x^5 - 21536*x^4 - 18172*x^3 + 51631*x^2 - 46426*x + 19321)
 
gp: K = bnfinit(x^20 - 6*x^19 + 18*x^18 - 35*x^17 + 3*x^16 - 8*x^15 + 443*x^14 - 1487*x^13 + 2989*x^12 - 167*x^11 - 6369*x^10 + 4885*x^9 + 19103*x^8 - 31548*x^7 + 7631*x^6 + 30774*x^5 - 21536*x^4 - 18172*x^3 + 51631*x^2 - 46426*x + 19321, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 18 x^{18} - 35 x^{17} + 3 x^{16} - 8 x^{15} + 443 x^{14} - 1487 x^{13} + 2989 x^{12} - 167 x^{11} - 6369 x^{10} + 4885 x^{9} + 19103 x^{8} - 31548 x^{7} + 7631 x^{6} + 30774 x^{5} - 21536 x^{4} - 18172 x^{3} + 51631 x^{2} - 46426 x + 19321 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(55374574082926437693343023378125=5^{5}\cdot 61^{10}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.65$
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:  $5, 61, 397$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is 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}{18327096600368963210466598669487534679785738781215} a^{19} - \frac{6508212703710373635339741415165469915485406558292}{18327096600368963210466598669487534679785738781215} a^{18} - \frac{1203568975040001144099878664228059801183066144266}{3665419320073792642093319733897506935957147756243} a^{17} - \frac{1807126556885254484109454324037911476669617952880}{3665419320073792642093319733897506935957147756243} a^{16} + \frac{2766146083835212599861808945356166537163182327618}{18327096600368963210466598669487534679785738781215} a^{15} - \frac{401447176434485869121996541885877896134213395822}{1409776661566843323882046051499041129214287598555} a^{14} + \frac{7430702631680228828388615268208742624830220012309}{18327096600368963210466598669487534679785738781215} a^{13} + \frac{7702393289372702685435977355040884395197698417729}{18327096600368963210466598669487534679785738781215} a^{12} + \frac{494056910086706719050959763192336397203902552419}{3665419320073792642093319733897506935957147756243} a^{11} + \frac{5129401001143575794780001908778922985412177348938}{18327096600368963210466598669487534679785738781215} a^{10} + \frac{6988282846438670602848737768350225075620047554428}{18327096600368963210466598669487534679785738781215} a^{9} + \frac{4934131005571607212764284073304992615969566487507}{18327096600368963210466598669487534679785738781215} a^{8} + \frac{6163061849086609554055162516671799207037896782986}{18327096600368963210466598669487534679785738781215} a^{7} - \frac{6760806958732304988705926245333741790681757820814}{18327096600368963210466598669487534679785738781215} a^{6} - \frac{1577644463208045612200456801177135420082910358869}{3665419320073792642093319733897506935957147756243} a^{5} + \frac{6089679595271337834180738816544653439052596003339}{18327096600368963210466598669487534679785738781215} a^{4} + \frac{216974271872083623767050821048594894084604230102}{3665419320073792642093319733897506935957147756243} a^{3} - \frac{7800817641609670586023544237076046752407634339037}{18327096600368963210466598669487534679785738781215} a^{2} + \frac{3193059133193208831588316973394843501504701265688}{18327096600368963210466598669487534679785738781215} a - \frac{43262910257210941050710420888797031192095592621}{131849615829992541082493515607823990502055674685}$

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

Class group and class number

$C_{16}$, which has order $16$ (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:  $9$
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:  \( 2930357.91985 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T174:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 960
The 35 conjugacy class representatives for t20n174
Character table for t20n174 is not computed

Intermediate fields

\(\Q(\sqrt{61}) \), 4.0.18605.1, 5.5.24217.1, 10.10.133115978404309.1

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 24 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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ R $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$61$61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
397Data not computed