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Properties

Label 20.0.43774679639...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 7^{10}$
Root discriminant $38.20$
Ramified primes $2, 3, 5, 7$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, 80, 144, 708, 1314, 3372, 4176, 6384, 6138, 8229, 2679, 3933, -798, 1269, -423, 255, -60, 17, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 17*x^18 - 60*x^17 + 255*x^16 - 423*x^15 + 1269*x^14 - 798*x^13 + 3933*x^12 + 2679*x^11 + 8229*x^10 + 6138*x^9 + 6384*x^8 + 4176*x^7 + 3372*x^6 + 1314*x^5 + 708*x^4 + 144*x^3 + 80*x^2 + 4)
 
gp: K = bnfinit(x^20 - 3*x^19 + 17*x^18 - 60*x^17 + 255*x^16 - 423*x^15 + 1269*x^14 - 798*x^13 + 3933*x^12 + 2679*x^11 + 8229*x^10 + 6138*x^9 + 6384*x^8 + 4176*x^7 + 3372*x^6 + 1314*x^5 + 708*x^4 + 144*x^3 + 80*x^2 + 4, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 17 x^{18} - 60 x^{17} + 255 x^{16} - 423 x^{15} + 1269 x^{14} - 798 x^{13} + 3933 x^{12} + 2679 x^{11} + 8229 x^{10} + 6138 x^{9} + 6384 x^{8} + 4176 x^{7} + 3372 x^{6} + 1314 x^{5} + 708 x^{4} + 144 x^{3} + 80 x^{2} + 4 \)

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:  \(43774679639190704400000000000000=2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.20$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{298} a^{17} - \frac{3}{149} a^{16} + \frac{3}{149} a^{15} + \frac{23}{298} a^{14} + \frac{3}{298} a^{13} + \frac{51}{149} a^{12} - \frac{29}{149} a^{11} + \frac{73}{298} a^{10} - \frac{113}{298} a^{9} - \frac{40}{149} a^{8} + \frac{5}{149} a^{7} - \frac{11}{298} a^{6} - \frac{51}{149} a^{5} + \frac{24}{149} a^{4} - \frac{7}{149} a^{3} - \frac{20}{149} a^{2} + \frac{57}{149} a - \frac{51}{149}$, $\frac{1}{1316266} a^{18} - \frac{2183}{1316266} a^{17} - \frac{208793}{1316266} a^{16} - \frac{11660}{658133} a^{15} - \frac{92189}{188038} a^{14} - \frac{70201}{1316266} a^{13} - \frac{99187}{1316266} a^{12} + \frac{196450}{658133} a^{11} - \frac{501883}{1316266} a^{10} - \frac{448419}{1316266} a^{9} - \frac{617467}{1316266} a^{8} - \frac{13827}{94019} a^{7} - \frac{237131}{658133} a^{6} + \frac{246343}{658133} a^{5} + \frac{172884}{658133} a^{4} + \frac{205641}{658133} a^{3} - \frac{195399}{658133} a^{2} - \frac{290573}{658133} a - \frac{302746}{658133}$, $\frac{1}{1010397948504693132943449422} a^{19} - \frac{120458347752475753605}{505198974252346566471724711} a^{18} - \frac{350726528106715540638008}{505198974252346566471724711} a^{17} + \frac{104168509876546083028894693}{505198974252346566471724711} a^{16} + \frac{60351174245185402212516179}{505198974252346566471724711} a^{15} - \frac{251685150930793296354863867}{1010397948504693132943449422} a^{14} - \frac{16264866844011309519817756}{505198974252346566471724711} a^{13} - \frac{128748936470633790981462143}{505198974252346566471724711} a^{12} + \frac{204878026536640178839179238}{505198974252346566471724711} a^{11} - \frac{134421632953375848295934521}{1010397948504693132943449422} a^{10} + \frac{212553077240634229066719280}{505198974252346566471724711} a^{9} + \frac{186632350838657918870107597}{505198974252346566471724711} a^{8} + \frac{30969287979808695353525131}{1010397948504693132943449422} a^{7} + \frac{312166298892819853245934395}{1010397948504693132943449422} a^{6} - \frac{30301766051727757718223439}{72171282036049509495960673} a^{5} - \frac{174150981545917423676781706}{505198974252346566471724711} a^{4} - \frac{245401434750063144181543423}{505198974252346566471724711} a^{3} - \frac{125016674265541234691042774}{505198974252346566471724711} a^{2} + \frac{233764332768364030044045791}{505198974252346566471724711} a - \frac{109035393326606620079316202}{505198974252346566471724711}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 85235565.63719633 \) (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

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-15}, \sqrt{21})\), 5.1.162000.1, 10.0.2205414540000000.2, 10.0.393660000000.1, 10.2.1323248724000000.1

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 R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{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$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{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}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$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}$