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Properties

Label 20.20.1201292780...8125.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{15}\cdot 7^{10}\cdot 139^{8}$
Root discriminant $63.68$
Ramified primes $5, 7, 139$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\times D_5$ (as 20T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-139, -27105, 344740, -1586360, 3317104, -2597056, -1533360, 4095435, -1892289, -1032083, 1183051, -135520, -215641, 75642, 11137, -8836, 431, 394, -49, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 49*x^18 + 394*x^17 + 431*x^16 - 8836*x^15 + 11137*x^14 + 75642*x^13 - 215641*x^12 - 135520*x^11 + 1183051*x^10 - 1032083*x^9 - 1892289*x^8 + 4095435*x^7 - 1533360*x^6 - 2597056*x^5 + 3317104*x^4 - 1586360*x^3 + 344740*x^2 - 27105*x - 139)
 
gp: K = bnfinit(x^20 - 6*x^19 - 49*x^18 + 394*x^17 + 431*x^16 - 8836*x^15 + 11137*x^14 + 75642*x^13 - 215641*x^12 - 135520*x^11 + 1183051*x^10 - 1032083*x^9 - 1892289*x^8 + 4095435*x^7 - 1533360*x^6 - 2597056*x^5 + 3317104*x^4 - 1586360*x^3 + 344740*x^2 - 27105*x - 139, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 49 x^{18} + 394 x^{17} + 431 x^{16} - 8836 x^{15} + 11137 x^{14} + 75642 x^{13} - 215641 x^{12} - 135520 x^{11} + 1183051 x^{10} - 1032083 x^{9} - 1892289 x^{8} + 4095435 x^{7} - 1533360 x^{6} - 2597056 x^{5} + 3317104 x^{4} - 1586360 x^{3} + 344740 x^{2} - 27105 x - 139 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1201292780904647323599596221923828125=5^{15}\cdot 7^{10}\cdot 139^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.68$
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:  $5, 7, 139$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{13} - \frac{1}{3} a^{11} - \frac{2}{9} a^{10} + \frac{4}{9} a^{9} - \frac{1}{3} a^{8} + \frac{2}{9} a^{6} + \frac{2}{9} a^{5} - \frac{4}{9} a^{4} + \frac{2}{9} a^{3} - \frac{2}{9} a^{2} + \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{14} - \frac{2}{9} a^{11} - \frac{2}{9} a^{10} - \frac{1}{3} a^{8} - \frac{4}{9} a^{7} - \frac{4}{9} a^{6} - \frac{4}{9} a^{5} + \frac{2}{9} a^{4} + \frac{1}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{15} + \frac{1}{9} a^{12} - \frac{2}{9} a^{11} + \frac{1}{3} a^{10} + \frac{2}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{2}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{3} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6792574991966730255577128509156668839} a^{19} - \frac{152600625115288180976353844900091665}{6792574991966730255577128509156668839} a^{18} + \frac{49225722459570807841185351089740622}{6792574991966730255577128509156668839} a^{17} - \frac{91200433829386518613787386446306956}{2264191663988910085192376169718889613} a^{16} + \frac{31601053130450502455471103943007746}{6792574991966730255577128509156668839} a^{15} - \frac{1075336760345791903836503606847911206}{6792574991966730255577128509156668839} a^{14} + \frac{1233004617724282172406310901416494}{754730554662970028397458723239629871} a^{13} - \frac{719049073661447555912426733995712340}{6792574991966730255577128509156668839} a^{12} - \frac{451604800788636824033620855904505671}{2264191663988910085192376169718889613} a^{11} + \frac{513794318530295699881118238708668701}{6792574991966730255577128509156668839} a^{10} - \frac{3084075559712911220971455144518497400}{6792574991966730255577128509156668839} a^{9} + \frac{1241845596499812347186651362007403793}{6792574991966730255577128509156668839} a^{8} - \frac{1427180689599680605201184025944142568}{6792574991966730255577128509156668839} a^{7} + \frac{128121348074832567771635769677667710}{754730554662970028397458723239629871} a^{6} - \frac{1273294909773158583661543879642379749}{6792574991966730255577128509156668839} a^{5} - \frac{1213144263817603573283493769597328}{15195917207979262316727356843750937} a^{4} - \frac{2126854554178316975086303908306721502}{6792574991966730255577128509156668839} a^{3} + \frac{822460832055478309184534575288120505}{2264191663988910085192376169718889613} a^{2} - \frac{947928242061209882835006489290006041}{6792574991966730255577128509156668839} a - \frac{2932569703270978681220643148133419765}{6792574991966730255577128509156668839}$

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:  $19$
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:  \( 386511180224 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times D_5$ (as 20T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 16 conjugacy class representatives for $C_4\times D_5$
Character table for $C_4\times D_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.6125.1, 5.5.23668225.1, 10.10.2800924373253125.1

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

Sibling fields

Galois closure: data not computed
Degree 20 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 $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ $20$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$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$$[\ ]_{4}^{2}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$139$$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$