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Properties

Label 20.4.25315355128...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 3^{10}\cdot 5^{15}\cdot 11^{8}$
Root discriminant $26.31$
Ramified primes $2, 3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_5:F_5$ (as 20T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, -25, -155, 44, 1273, 2635, 2795, 1956, -11, -1801, -865, 874, 579, -277, -183, 66, 33, -11, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 11*x^18 + 33*x^17 + 66*x^16 - 183*x^15 - 277*x^14 + 579*x^13 + 874*x^12 - 865*x^11 - 1801*x^10 - 11*x^9 + 1956*x^8 + 2795*x^7 + 2635*x^6 + 1273*x^5 + 44*x^4 - 155*x^3 - 25*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 - 11*x^18 + 33*x^17 + 66*x^16 - 183*x^15 - 277*x^14 + 579*x^13 + 874*x^12 - 865*x^11 - 1801*x^10 - 11*x^9 + 1956*x^8 + 2795*x^7 + 2635*x^6 + 1273*x^5 + 44*x^4 - 155*x^3 - 25*x^2 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 11 x^{18} + 33 x^{17} + 66 x^{16} - 183 x^{15} - 277 x^{14} + 579 x^{13} + 874 x^{12} - 865 x^{11} - 1801 x^{10} - 11 x^{9} + 1956 x^{8} + 2795 x^{7} + 2635 x^{6} + 1273 x^{5} + 44 x^{4} - 155 x^{3} - 25 x^{2} + 5 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(25315355128338000000000000000=2^{16}\cdot 3^{10}\cdot 5^{15}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.31$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{70425895069103692002259} a^{19} - \frac{11776493487270930919719}{70425895069103692002259} a^{18} - \frac{25072067961645386411173}{70425895069103692002259} a^{17} + \frac{5085792286891520167884}{70425895069103692002259} a^{16} - \frac{21563366434184108390482}{70425895069103692002259} a^{15} + \frac{12664609801207314831493}{70425895069103692002259} a^{14} + \frac{32829399173932500527017}{70425895069103692002259} a^{13} + \frac{30652549084932305846497}{70425895069103692002259} a^{12} + \frac{24227186871133206480626}{70425895069103692002259} a^{11} + \frac{19254743738955505947647}{70425895069103692002259} a^{10} + \frac{8251199796130059114437}{70425895069103692002259} a^{9} - \frac{10397389622961396173746}{70425895069103692002259} a^{8} - \frac{27626793903964750758679}{70425895069103692002259} a^{7} - \frac{8349286581544808820559}{70425895069103692002259} a^{6} + \frac{2221606700001259285503}{70425895069103692002259} a^{5} + \frac{10211124345711735763596}{70425895069103692002259} a^{4} + \frac{18025910483498127151781}{70425895069103692002259} a^{3} - \frac{5586760522520603009734}{70425895069103692002259} a^{2} + \frac{18505079738713330868986}{70425895069103692002259} a - \frac{32627016069782205573353}{70425895069103692002259}$

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:  $11$
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:  \( 2156441.45048 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_5:F_5$ (as 20T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 20 conjugacy class representatives for $C_2\times C_5:F_5$
Character table for $C_2\times C_5:F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 10.2.292820000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.3.1$x^{4} - 5$$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}$
$11$11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11.10.8.3$x^{10} - 11 x^{5} + 847$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$