↑

Properties

Label 20.0.24032520061...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{23}\cdot 17^{10}$
Root discriminant $26.24$
Ramified primes $5, 17$
Class number $2$
Class group $[2]$
Galois group $F_5$ (as 20T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14741, 31085, 71185, 1020, 54230, -32983, 28430, -25850, 16915, -12320, 8584, -5715, 3375, -2005, 1015, -482, 200, -65, 20, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 20*x^18 - 65*x^17 + 200*x^16 - 482*x^15 + 1015*x^14 - 2005*x^13 + 3375*x^12 - 5715*x^11 + 8584*x^10 - 12320*x^9 + 16915*x^8 - 25850*x^7 + 28430*x^6 - 32983*x^5 + 54230*x^4 + 1020*x^3 + 71185*x^2 + 31085*x + 14741)
 
gp: K = bnfinit(x^20 - 5*x^19 + 20*x^18 - 65*x^17 + 200*x^16 - 482*x^15 + 1015*x^14 - 2005*x^13 + 3375*x^12 - 5715*x^11 + 8584*x^10 - 12320*x^9 + 16915*x^8 - 25850*x^7 + 28430*x^6 - 32983*x^5 + 54230*x^4 + 1020*x^3 + 71185*x^2 + 31085*x + 14741, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 20 x^{18} - 65 x^{17} + 200 x^{16} - 482 x^{15} + 1015 x^{14} - 2005 x^{13} + 3375 x^{12} - 5715 x^{11} + 8584 x^{10} - 12320 x^{9} + 16915 x^{8} - 25850 x^{7} + 28430 x^{6} - 32983 x^{5} + 54230 x^{4} + 1020 x^{3} + 71185 x^{2} + 31085 x + 14741 \)

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:  \(24032520061123371124267578125=5^{23}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.24$
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:  $5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{10} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{12} a^{16} - \frac{1}{12} a^{14} - \frac{1}{12} a^{13} + \frac{1}{12} a^{12} - \frac{1}{4} a^{10} + \frac{1}{12} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{6} a^{6} - \frac{5}{12} a^{5} + \frac{1}{4} a^{4} - \frac{5}{12} a^{3} - \frac{1}{12} a^{2} - \frac{5}{12} a - \frac{5}{12}$, $\frac{1}{12} a^{17} - \frac{1}{12} a^{15} - \frac{1}{12} a^{14} + \frac{1}{12} a^{13} - \frac{1}{4} a^{11} + \frac{1}{12} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{6} a^{7} - \frac{5}{12} a^{6} + \frac{1}{4} a^{5} - \frac{5}{12} a^{4} - \frac{1}{12} a^{3} - \frac{5}{12} a^{2} - \frac{5}{12} a$, $\frac{1}{1293072} a^{18} - \frac{46247}{1293072} a^{17} + \frac{3747}{431024} a^{16} - \frac{421}{7347} a^{15} - \frac{80437}{1293072} a^{14} - \frac{6487}{107756} a^{13} - \frac{38567}{323268} a^{12} + \frac{249655}{1293072} a^{11} - \frac{213575}{1293072} a^{10} - \frac{1877}{14694} a^{9} - \frac{931}{117552} a^{8} + \frac{1043}{58776} a^{7} - \frac{35887}{80817} a^{6} - \frac{78395}{161634} a^{5} + \frac{44815}{215512} a^{4} + \frac{60949}{1293072} a^{3} - \frac{34675}{646536} a^{2} + \frac{7181}{41712} a - \frac{394127}{1293072}$, $\frac{1}{19646785863110168860922547228616072512} a^{19} + \frac{998513483590752582230716487047}{9823392931555084430461273614308036256} a^{18} - \frac{12668176495129279632331959812797711}{316883642953389820337460439171226976} a^{17} + \frac{57349800536460378462173660228793131}{1786071442100924441902049748056006592} a^{16} - \frac{413310082475749001842780492782209679}{6548928621036722953640849076205357504} a^{15} + \frac{1189342195031174692466952144990496199}{19646785863110168860922547228616072512} a^{14} + \frac{119468873213068033430916505955863723}{4911696465777542215230636807154018128} a^{13} - \frac{2981425902820364327369885562469033537}{19646785863110168860922547228616072512} a^{12} - \frac{1196387926295278108819715580667443373}{4911696465777542215230636807154018128} a^{11} + \frac{4204153352089079104780335367006963}{1786071442100924441902049748056006592} a^{10} - \frac{350990439401021994203449996737526607}{1786071442100924441902049748056006592} a^{9} - \frac{27900174606624422017908653838624431}{595357147366974813967349916018668864} a^{8} - \frac{1390691069236268837689397365772250061}{9823392931555084430461273614308036256} a^{7} - \frac{1141976273619128185847956655665600615}{2455848232888771107615318403577009064} a^{6} + \frac{1050848336567653505706262678675780051}{9823392931555084430461273614308036256} a^{5} - \frac{9684289283137369516833699739676490293}{19646785863110168860922547228616072512} a^{4} + \frac{5228948915304901803276673130338793111}{19646785863110168860922547228616072512} a^{3} + \frac{2066961421502551532143086283195382123}{6548928621036722953640849076205357504} a^{2} + \frac{534161910394590311805606933370568455}{1637232155259180738410212269051339376} a + \frac{500809033514916920289877854254575787}{1786071442100924441902049748056006592}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 382451.294806 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.36125.1, 5.1.903125.1 x5, 10.2.4078173828125.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.903125.1
Degree 10 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
5Data not computed
$17$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$