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Properties

Label 20.0.68634231647...3013.1
Degree $20$
Signature $[0, 10]$
Discriminant $7^{15}\cdot 11^{9}\cdot 19^{10}$
Root discriminant $55.19$
Ramified primes $7, 11, 19$
Class number Not computed
Class group Not computed
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![964658227471, -399048852098, 106836459007, -25819961020, 23164696201, -1350280281, 101111009, -13194435, 46160552, 10604317, 1940277, -126135, 225499, -14490, -3202, -3504, -242, 72, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 72*x^17 - 242*x^16 - 3504*x^15 - 3202*x^14 - 14490*x^13 + 225499*x^12 - 126135*x^11 + 1940277*x^10 + 10604317*x^9 + 46160552*x^8 - 13194435*x^7 + 101111009*x^6 - 1350280281*x^5 + 23164696201*x^4 - 25819961020*x^3 + 106836459007*x^2 - 399048852098*x + 964658227471)
 
gp: K = bnfinit(x^20 - 2*x^19 + 72*x^17 - 242*x^16 - 3504*x^15 - 3202*x^14 - 14490*x^13 + 225499*x^12 - 126135*x^11 + 1940277*x^10 + 10604317*x^9 + 46160552*x^8 - 13194435*x^7 + 101111009*x^6 - 1350280281*x^5 + 23164696201*x^4 - 25819961020*x^3 + 106836459007*x^2 - 399048852098*x + 964658227471, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 72 x^{17} - 242 x^{16} - 3504 x^{15} - 3202 x^{14} - 14490 x^{13} + 225499 x^{12} - 126135 x^{11} + 1940277 x^{10} + 10604317 x^{9} + 46160552 x^{8} - 13194435 x^{7} + 101111009 x^{6} - 1350280281 x^{5} + 23164696201 x^{4} - 25819961020 x^{3} + 106836459007 x^{2} - 399048852098 x + 964658227471 \)

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:  \(68634231647302196434922446986223013=7^{15}\cdot 11^{9}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.19$
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:  $7, 11, 19$
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}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a^{19} + \frac{1474304401132635727492669139881752260294380724125641801365051626834745396047178313803533840956538043248743458101692}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a^{18} + \frac{1396635873842625438418808912645116905939814611474881643734568476995607457563182397702318500522417398156446539025645}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a^{17} + \frac{1272180124380654885911286514820196967985639342391485682053684402540805179472708242885366671655363505501470286648884}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a^{16} + \frac{2031234820558825677843013493986836485935992667330091265998185950927466383132009335758678225958655010965145499611462}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a^{15} + \frac{1509646264334235525189588487810582428945943779921547919937419342744213966569358776522883622563273753900936164983086}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a^{14} + \frac{893940462938894508593709424498301799088536912399882544354397469849432378438770224744765687356478918818216439290479}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a^{13} - \frac{148366306439606049780929450868044563561824647223118786170665556517839308195460570285239787546091309944073773243305}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a^{12} - \frac{1889775346721251376983903595542142796265789260733831601183856090844198373549764367060119168581959703903825358691470}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a^{11} - \frac{1360313366589466043488478337117678148126744646176793928954785753080774548980087586922236326331015078901958072017868}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a^{10} + \frac{1205934412025192875734041222174408538164537537807021645905078896388403964352404239633813357575400065772201039291128}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a^{9} + \frac{2078344890636912567155785784970858047192982457144645974295223826222626163252370178685229619336682400778305334454512}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a^{8} + \frac{2316522117971107612644665202685196932297278345637542481755923039399473630304433397055770713670573881667259612591292}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a^{7} + \frac{1079191475547479654831043302430077991922475251798931090898676883223980576189611177437351657200030675502197104430549}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a^{6} - \frac{2517260284040177493003097315745290782499882347733114279360897349758474369374892995492610392264705586103344843662163}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a^{5} + \frac{2051862540704813027675564968492587249840615970105808910178689865680073560054934516158583397380975292163283163185326}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a^{4} + \frac{2312942735477347315519304681898556055662893204529760862240635762040306069608882356692233023195791555353940831094322}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a^{3} + \frac{2327236100432153527595159581578713354527947341641080457433228303376328545334239050226469672747240702102617664762344}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a^{2} - \frac{716693571496968265660706674230934700815358714514810823491731519350483875904163777784908492819823841164466454298305}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101} a - \frac{1136212562919725883796475935850320054351405875382929932304414647844912323835463564772539777604878210954412597828338}{5105877969365029395745385465270665036054615795974921360839657866312612596002548648067588160507611654623437400674101}$

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:  $9$
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:  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

$C_5\times C_5:D_4$ (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

\(\Q(\sqrt{-7}) \), 4.0.1362053.2, 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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ $20$ $20$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{10}$ $20$

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
7Data not computed
$11$11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$
19Data not computed