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Properties

Label 20.0.96780543004...7717.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 7^{15}\cdot 11^{13}$
Root discriminant $35.42$
Ramified primes $3, 7, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![103856419, -53651341, 173029525, -62702972, 119110990, -30553298, 46563112, -8666349, 11918963, -1737644, 2138211, -266297, 271844, -29499, 24203, -2178, 1454, -97, 54, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 54*x^18 - 97*x^17 + 1454*x^16 - 2178*x^15 + 24203*x^14 - 29499*x^13 + 271844*x^12 - 266297*x^11 + 2138211*x^10 - 1737644*x^9 + 11918963*x^8 - 8666349*x^7 + 46563112*x^6 - 30553298*x^5 + 119110990*x^4 - 62702972*x^3 + 173029525*x^2 - 53651341*x + 103856419)
 
gp: K = bnfinit(x^20 - 2*x^19 + 54*x^18 - 97*x^17 + 1454*x^16 - 2178*x^15 + 24203*x^14 - 29499*x^13 + 271844*x^12 - 266297*x^11 + 2138211*x^10 - 1737644*x^9 + 11918963*x^8 - 8666349*x^7 + 46563112*x^6 - 30553298*x^5 + 119110990*x^4 - 62702972*x^3 + 173029525*x^2 - 53651341*x + 103856419, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 54 x^{18} - 97 x^{17} + 1454 x^{16} - 2178 x^{15} + 24203 x^{14} - 29499 x^{13} + 271844 x^{12} - 266297 x^{11} + 2138211 x^{10} - 1737644 x^{9} + 11918963 x^{8} - 8666349 x^{7} + 46563112 x^{6} - 30553298 x^{5} + 119110990 x^{4} - 62702972 x^{3} + 173029525 x^{2} - 53651341 x + 103856419 \)

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:  \(9678054300479022526566826737717=3^{10}\cdot 7^{15}\cdot 11^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.42$
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:  $3, 7, 11$
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}{2188459369635275291342227254564356159819140518458111509016424103079159} a^{19} - \frac{63367682482634117131536896710777015538679754844169296110353571297411}{2188459369635275291342227254564356159819140518458111509016424103079159} a^{18} + \frac{591727772454637601936293995769408691922120270961466763863057179980849}{2188459369635275291342227254564356159819140518458111509016424103079159} a^{17} + \frac{237320797307247655925528566945972028983496104601853921172709515565200}{2188459369635275291342227254564356159819140518458111509016424103079159} a^{16} + \frac{817995803188288222188217327143527590402464765117548138800611684706163}{2188459369635275291342227254564356159819140518458111509016424103079159} a^{15} + \frac{1029053173262068588696792741325391836035510731114275682432350406654336}{2188459369635275291342227254564356159819140518458111509016424103079159} a^{14} + \frac{28183738819150229835718328200990114432525555328893062928307220254056}{2188459369635275291342227254564356159819140518458111509016424103079159} a^{13} - \frac{394882289125310988228442779419886588009521620836919287828320797513765}{2188459369635275291342227254564356159819140518458111509016424103079159} a^{12} + \frac{677882400767370036191205455472778828814488790090624404306169521766277}{2188459369635275291342227254564356159819140518458111509016424103079159} a^{11} + \frac{72194481248495025111635432006446844663508675269513822654410336346709}{2188459369635275291342227254564356159819140518458111509016424103079159} a^{10} - \frac{660456890271996678092183247238261700716886006577439664994585130578066}{2188459369635275291342227254564356159819140518458111509016424103079159} a^{9} - \frac{754575026197457654243249873136866195713709199492045633783833750461292}{2188459369635275291342227254564356159819140518458111509016424103079159} a^{8} + \frac{1005453776316832888693591514019996349684338067938710969290265770527607}{2188459369635275291342227254564356159819140518458111509016424103079159} a^{7} - \frac{1036374412267173903254217011529692422016471066159955286299577453370749}{2188459369635275291342227254564356159819140518458111509016424103079159} a^{6} - \frac{108326704434066174812195870893284550363455915578949049473568554869114}{2188459369635275291342227254564356159819140518458111509016424103079159} a^{5} - \frac{209462726999804289182790330172896710419096995265641482354927768632223}{2188459369635275291342227254564356159819140518458111509016424103079159} a^{4} - \frac{604619404287966822309819333854182881385114139354445167709920284917364}{2188459369635275291342227254564356159819140518458111509016424103079159} a^{3} + \frac{111497192709205084285214578802183361066615332706727588882886159767877}{2188459369635275291342227254564356159819140518458111509016424103079159} a^{2} - \frac{976121882349822309357881056316977592930256624676561589459748882764102}{2188459369635275291342227254564356159819140518458111509016424103079159} a - \frac{1010799837690829528880652143347473115952415081110392409785750452630409}{2188459369635275291342227254564356159819140518458111509016424103079159}$

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:  \( 7036306.946137199 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.33957.1, 10.0.246071287.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ R $20$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{10}$ $20$

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
3Data not computed
7Data not computed
$11$11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$