↑

Properties

Label 20.0.64197916927...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 23^{10}$
Root discriminant $69.24$
Ramified primes $2, 3, 5, 23$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![56607784, 59948856, 55936280, 49085616, 47245728, 17022462, 22362606, 4667544, 6623538, 789438, 1474509, 52512, 231729, -756, 23886, -276, 1560, -12, 59, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 59*x^18 - 12*x^17 + 1560*x^16 - 276*x^15 + 23886*x^14 - 756*x^13 + 231729*x^12 + 52512*x^11 + 1474509*x^10 + 789438*x^9 + 6623538*x^8 + 4667544*x^7 + 22362606*x^6 + 17022462*x^5 + 47245728*x^4 + 49085616*x^3 + 55936280*x^2 + 59948856*x + 56607784)
 
gp: K = bnfinit(x^20 + 59*x^18 - 12*x^17 + 1560*x^16 - 276*x^15 + 23886*x^14 - 756*x^13 + 231729*x^12 + 52512*x^11 + 1474509*x^10 + 789438*x^9 + 6623538*x^8 + 4667544*x^7 + 22362606*x^6 + 17022462*x^5 + 47245728*x^4 + 49085616*x^3 + 55936280*x^2 + 59948856*x + 56607784, 1)
 

Normalized defining polynomial

\( x^{20} + 59 x^{18} - 12 x^{17} + 1560 x^{16} - 276 x^{15} + 23886 x^{14} - 756 x^{13} + 231729 x^{12} + 52512 x^{11} + 1474509 x^{10} + 789438 x^{9} + 6623538 x^{8} + 4667544 x^{7} + 22362606 x^{6} + 17022462 x^{5} + 47245728 x^{4} + 49085616 x^{3} + 55936280 x^{2} + 59948856 x + 56607784 \)

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:  \(6419791692782351621744400000000000000=2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.24$
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:  $2, 3, 5, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{6} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{13} - \frac{1}{3} a^{7} - \frac{1}{2} a^{5} - \frac{1}{3} a$, $\frac{1}{18} a^{14} - \frac{1}{18} a^{12} - \frac{1}{6} a^{9} + \frac{1}{18} a^{8} - \frac{1}{3} a^{7} - \frac{1}{18} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{3} - \frac{1}{9} a^{2} + \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{18} a^{15} - \frac{1}{18} a^{13} - \frac{1}{9} a^{9} - \frac{7}{18} a^{7} - \frac{1}{2} a^{5} - \frac{4}{9} a^{3} + \frac{4}{9} a$, $\frac{1}{180} a^{16} + \frac{1}{45} a^{15} - \frac{1}{18} a^{13} + \frac{7}{90} a^{12} - \frac{1}{30} a^{11} - \frac{2}{45} a^{10} + \frac{1}{18} a^{9} + \frac{7}{60} a^{8} - \frac{23}{90} a^{7} + \frac{1}{90} a^{6} + \frac{13}{30} a^{5} + \frac{17}{90} a^{4} + \frac{29}{90} a^{3} - \frac{7}{15} a^{2} - \frac{22}{45} a + \frac{14}{45}$, $\frac{1}{180} a^{17} + \frac{1}{45} a^{15} + \frac{1}{45} a^{13} - \frac{1}{15} a^{12} - \frac{7}{90} a^{11} + \frac{1}{15} a^{10} - \frac{29}{180} a^{9} + \frac{4}{45} a^{7} + \frac{1}{6} a^{6} + \frac{11}{90} a^{5} + \frac{7}{30} a^{4} - \frac{14}{45} a^{3} - \frac{2}{5} a^{2} + \frac{22}{45} a + \frac{1}{5}$, $\frac{1}{180} a^{18} + \frac{1}{45} a^{15} + \frac{1}{45} a^{14} + \frac{2}{45} a^{13} - \frac{1}{18} a^{12} + \frac{1}{30} a^{11} + \frac{1}{60} a^{10} + \frac{1}{18} a^{9} - \frac{2}{45} a^{8} - \frac{19}{45} a^{7} + \frac{7}{90} a^{6} - \frac{1}{3} a^{5} - \frac{1}{15} a^{4} + \frac{19}{45} a^{3} + \frac{1}{45} a^{2} + \frac{17}{45} a + \frac{19}{45}$, $\frac{1}{2934660225643079172518730087406255411218231285927481012860} a^{19} - \frac{4377357491283066914899017609468844374503002569370225171}{2934660225643079172518730087406255411218231285927481012860} a^{18} + \frac{460474763929381880996090247732308840271622446308863909}{293466022564307917251873008740625541121823128592748101286} a^{17} - \frac{69629785802073882069543633320691818551962015764096773}{48911003760717986208645501456770923520303854765458016881} a^{16} + \frac{37564260132386193538203802847008156813418501750516106667}{1467330112821539586259365043703127705609115642963740506430} a^{15} - \frac{11286273464163471632742007043649067653128152377831478939}{733665056410769793129682521851563852804557821481870253215} a^{14} - \frac{6195126567323831352992541814456959280917367485394005968}{81518339601196643681075835761284872533839757942430028135} a^{13} - \frac{23127009960754410459706747410374145831719650544065242647}{293466022564307917251873008740625541121823128592748101286} a^{12} + \frac{5278185024746260839044141416721472515133083332055742103}{419237175091868453216961441058036487316890183703925858980} a^{11} + \frac{34197939724864505918357672867045022855436076954321471303}{978220075214359724172910029135418470406077095309160337620} a^{10} + \frac{243958234146389113206516318978938274593003794605996790951}{1467330112821539586259365043703127705609115642963740506430} a^{9} + \frac{47259501182786275194338005587035728365684632433404629759}{1467330112821539586259365043703127705609115642963740506430} a^{8} - \frac{78038744832389616210694254698054938100962717052577276143}{244555018803589931043227507283854617601519273827290084405} a^{7} - \frac{326875023071080546325865895980742643991835851646711313761}{1467330112821539586259365043703127705609115642963740506430} a^{6} - \frac{61261346865823145274737277956157957032870152497905402741}{733665056410769793129682521851563852804557821481870253215} a^{5} + \frac{5009427887162119501263254833436000133454568157567912366}{11645477085885234811582262251612124647691393991775718305} a^{4} + \frac{234261119522268407706137398798133978210456220587529697874}{733665056410769793129682521851563852804557821481870253215} a^{3} + \frac{57303952779007773101511002808008923167237361170506791333}{146733011282153958625936504370312770560911564296374050643} a^{2} - \frac{6612415165464390542445495433889203615079255822806899923}{20961858754593422660848072052901824365844509185196292949} a + \frac{25140252631448508191311961652386047539191622336037214575}{146733011282153958625936504370312770560911564296374050643}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 54130298678.98963 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

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 $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{345}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-15}, \sqrt{-23})\), 5.1.162000.1, 10.0.393660000000.1, 10.2.2533730785380000000.1, 10.0.168915385692000000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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}$
$3$3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$