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Properties

Label 20.4.27326167244...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 5^{22}\cdot 53^{10}$
Root discriminant $74.44$
Ramified primes $2, 5, 53$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![288273254476, 2153019700, -185154483860, -360381720, 54615375960, -6751886, -9777028020, 5771750, 1177877180, -550380, -99761943, 22530, 6010785, -290, -254260, -4, 7230, 0, -125, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 125*x^18 + 7230*x^16 - 4*x^15 - 254260*x^14 - 290*x^13 + 6010785*x^12 + 22530*x^11 - 99761943*x^10 - 550380*x^9 + 1177877180*x^8 + 5771750*x^7 - 9777028020*x^6 - 6751886*x^5 + 54615375960*x^4 - 360381720*x^3 - 185154483860*x^2 + 2153019700*x + 288273254476)
 
gp: K = bnfinit(x^20 - 125*x^18 + 7230*x^16 - 4*x^15 - 254260*x^14 - 290*x^13 + 6010785*x^12 + 22530*x^11 - 99761943*x^10 - 550380*x^9 + 1177877180*x^8 + 5771750*x^7 - 9777028020*x^6 - 6751886*x^5 + 54615375960*x^4 - 360381720*x^3 - 185154483860*x^2 + 2153019700*x + 288273254476, 1)
 

Normalized defining polynomial

\( x^{20} - 125 x^{18} + 7230 x^{16} - 4 x^{15} - 254260 x^{14} - 290 x^{13} + 6010785 x^{12} + 22530 x^{11} - 99761943 x^{10} - 550380 x^{9} + 1177877180 x^{8} + 5771750 x^{7} - 9777028020 x^{6} - 6751886 x^{5} + 54615375960 x^{4} - 360381720 x^{3} - 185154483860 x^{2} + 2153019700 x + 288273254476 \)

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:  \(27326167244611413906250000000000000000=2^{16}\cdot 5^{22}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.44$
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:  $2, 5, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9}$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} + \frac{1}{7} a^{15} - \frac{3}{14} a^{13} + \frac{3}{14} a^{12} - \frac{3}{14} a^{11} - \frac{3}{14} a^{10} + \frac{1}{14} a^{9} + \frac{5}{14} a^{8} + \frac{5}{14} a^{7} + \frac{1}{7} a^{6} + \frac{5}{14} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{21176055205980755947948769873282362668812663413876822690432984521069319854198} a^{19} - \frac{408813465123210452689123147077643772553922631455519807837715352074240421581}{21176055205980755947948769873282362668812663413876822690432984521069319854198} a^{18} + \frac{553226029562035686240611403938253306484411708290140859321379678315958557925}{3025150743711536563992681410468908952687523344839546098633283503009902836314} a^{17} + \frac{1733136213869390801591827359192662393478911072133202945407461443773668630497}{10588027602990377973974384936641181334406331706938411345216492260534659927099} a^{16} + \frac{186360597516179920448501764999597184461180785280936473016155287550410732825}{3025150743711536563992681410468908952687523344839546098633283503009902836314} a^{15} + \frac{1803747055036439393271820430265079266931436823041644162789319922706925149592}{10588027602990377973974384936641181334406331706938411345216492260534659927099} a^{14} + \frac{865713212573038080781555851618304995899575078555462584055581884289597681735}{3529342534330125991324794978880393778135443902312803781738830753511553309033} a^{13} - \frac{1262667463804552009405952011713291575750531967847128165109749038933968079728}{10588027602990377973974384936641181334406331706938411345216492260534659927099} a^{12} + \frac{4201260572677112284287184564198125395184963945458863853837866678428974925115}{21176055205980755947948769873282362668812663413876822690432984521069319854198} a^{11} + \frac{3224447219297052907113717409988652589220815004192741817479534683468681355203}{21176055205980755947948769873282362668812663413876822690432984521069319854198} a^{10} + \frac{8999312971120373756699498757239313668198403969335009609573660632764129398959}{21176055205980755947948769873282362668812663413876822690432984521069319854198} a^{9} + \frac{4911124346224959583098136838357007883244499555844334487916783510843960281041}{10588027602990377973974384936641181334406331706938411345216492260534659927099} a^{8} + \frac{3302914687304793896626223428585839683887102078755473588340068422802046826103}{7058685068660251982649589957760787556270887804625607563477661507023106618066} a^{7} - \frac{1858328493408411561866724626946498608831832570372116846928388940733561822047}{10588027602990377973974384936641181334406331706938411345216492260534659927099} a^{6} - \frac{3641568509012956685438848764672747022199420493624140364625941559215427437701}{10588027602990377973974384936641181334406331706938411345216492260534659927099} a^{5} + \frac{353273441468187898164991204802902292578228428881926972110277629698471720179}{3529342534330125991324794978880393778135443902312803781738830753511553309033} a^{4} - \frac{995419916967626130158227099343643243110296594791664070955918555479716377624}{3529342534330125991324794978880393778135443902312803781738830753511553309033} a^{3} - \frac{718967643038849588443709225914567167587872429344939965211045595642399187697}{3529342534330125991324794978880393778135443902312803781738830753511553309033} a^{2} - \frac{2827956102167413461440565039237826341188075395273900784978207423189355076096}{10588027602990377973974384936641181334406331706938411345216492260534659927099} a + \frac{38963861011210592998413157376982087721619123651420156449549103857521961165}{1512575371855768281996340705234454476343761672419773049316641751504951418157}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{53}) \), \(\Q(\sqrt{265}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{53})\), 5.1.50000.1, 10.2.5227443662500000000.1, 10.2.1045488732500000000.1, 10.2.12500000000.1

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 10 siblings: 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$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}$
5Data not computed
$53$53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.8.4.1$x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
53.8.4.1$x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$