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Properties

Label 20.0.27571698369...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{15}\cdot 13^{10}$
Root discriminant $20.99$
Ramified primes $2, 5, 13$
Class number $2$
Class group $[2]$
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 50, 190, 205, 161, 66, -58, 664, 724, 946, 758, 106, 553, -206, 358, -153, 110, -32, 16, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 16*x^18 - 32*x^17 + 110*x^16 - 153*x^15 + 358*x^14 - 206*x^13 + 553*x^12 + 106*x^11 + 758*x^10 + 946*x^9 + 724*x^8 + 664*x^7 - 58*x^6 + 66*x^5 + 161*x^4 + 205*x^3 + 190*x^2 + 50*x + 25)
 
gp: K = bnfinit(x^20 - 3*x^19 + 16*x^18 - 32*x^17 + 110*x^16 - 153*x^15 + 358*x^14 - 206*x^13 + 553*x^12 + 106*x^11 + 758*x^10 + 946*x^9 + 724*x^8 + 664*x^7 - 58*x^6 + 66*x^5 + 161*x^4 + 205*x^3 + 190*x^2 + 50*x + 25, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 16 x^{18} - 32 x^{17} + 110 x^{16} - 153 x^{15} + 358 x^{14} - 206 x^{13} + 553 x^{12} + 106 x^{11} + 758 x^{10} + 946 x^{9} + 724 x^{8} + 664 x^{7} - 58 x^{6} + 66 x^{5} + 161 x^{4} + 205 x^{3} + 190 x^{2} + 50 x + 25 \)

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:  \(275716983698000000000000000=2^{16}\cdot 5^{15}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.99$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{13} + \frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{13} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{4} + \frac{1}{5} a^{2}$, $\frac{1}{15} a^{16} - \frac{2}{5} a^{13} + \frac{1}{15} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{9} + \frac{1}{15} a^{7} - \frac{4}{15} a^{6} - \frac{2}{5} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{4}{15} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{15} a^{17} - \frac{1}{3} a^{13} + \frac{2}{5} a^{12} - \frac{1}{3} a^{8} - \frac{4}{15} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{15} a^{2} - \frac{1}{3} a$, $\frac{1}{825} a^{18} + \frac{6}{275} a^{17} - \frac{1}{33} a^{16} - \frac{23}{275} a^{15} + \frac{61}{825} a^{14} + \frac{113}{275} a^{13} - \frac{262}{825} a^{12} - \frac{98}{275} a^{11} - \frac{96}{275} a^{10} + \frac{94}{825} a^{9} + \frac{149}{825} a^{8} + \frac{164}{825} a^{7} + \frac{92}{825} a^{6} - \frac{13}{165} a^{5} + \frac{124}{825} a^{4} - \frac{23}{55} a^{3} - \frac{38}{165} a^{2} - \frac{10}{33} a - \frac{10}{33}$, $\frac{1}{417958958729132720153925} a^{19} + \frac{19717085007349435296}{139319652909710906717975} a^{18} - \frac{236353171379196704958}{27863930581942181343595} a^{17} + \frac{12744644379293607045596}{417958958729132720153925} a^{16} - \frac{34858329710354638115189}{417958958729132720153925} a^{15} - \frac{5622658238756447064832}{139319652909710906717975} a^{14} + \frac{1452669547578471816746}{4494182351926158281225} a^{13} + \frac{133835542224862540926626}{417958958729132720153925} a^{12} - \frac{12425928417592721114591}{139319652909710906717975} a^{11} - \frac{22454058784373324832191}{417958958729132720153925} a^{10} - \frac{50836330315818792509866}{417958958729132720153925} a^{9} + \frac{30035958519028410594423}{139319652909710906717975} a^{8} + \frac{37980413229739492771779}{139319652909710906717975} a^{7} + \frac{7935898093297362192658}{83591791745826544030785} a^{6} + \frac{1564090278308497256084}{417958958729132720153925} a^{5} + \frac{2273998223420117452322}{27863930581942181343595} a^{4} - \frac{6372783117414498835219}{27863930581942181343595} a^{3} - \frac{3859872942465488069278}{83591791745826544030785} a^{2} + \frac{1553492492008356870725}{16718358349165308806157} a - \frac{1004892135758930919175}{16718358349165308806157}$

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:  \( 72830.9231902 \)
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

\(\Q(\sqrt{5}) \), 4.0.21125.1, 5.1.338000.1 x5, 10.2.571220000000.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.338000.1
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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$