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Properties

Label 20.4.680...129.4
Degree $20$
Signature $[4, 8]$
Discriminant $6.802\times 10^{24}$
Root discriminant \(17.44\)
Ramified primes $11,23$
Class number $1$
Class group trivial
Galois group $C_2\wr C_5$ (as 20T40)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 5*x^18 + 10*x^17 + 25*x^16 - 52*x^15 - 76*x^14 + 234*x^13 - 11*x^12 - 464*x^11 + 425*x^10 + 206*x^9 - 572*x^8 + 336*x^7 + 3*x^6 - 168*x^5 + 243*x^4 - 204*x^3 + 87*x^2 - 16*x + 1)
 
gp: K = bnfinit(y^20 - 2*y^19 - 5*y^18 + 10*y^17 + 25*y^16 - 52*y^15 - 76*y^14 + 234*y^13 - 11*y^12 - 464*y^11 + 425*y^10 + 206*y^9 - 572*y^8 + 336*y^7 + 3*y^6 - 168*y^5 + 243*y^4 - 204*y^3 + 87*y^2 - 16*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 - 5*x^18 + 10*x^17 + 25*x^16 - 52*x^15 - 76*x^14 + 234*x^13 - 11*x^12 - 464*x^11 + 425*x^10 + 206*x^9 - 572*x^8 + 336*x^7 + 3*x^6 - 168*x^5 + 243*x^4 - 204*x^3 + 87*x^2 - 16*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 - 5*x^18 + 10*x^17 + 25*x^16 - 52*x^15 - 76*x^14 + 234*x^13 - 11*x^12 - 464*x^11 + 425*x^10 + 206*x^9 - 572*x^8 + 336*x^7 + 3*x^6 - 168*x^5 + 243*x^4 - 204*x^3 + 87*x^2 - 16*x + 1)
 

\( x^{20} - 2 x^{19} - 5 x^{18} + 10 x^{17} + 25 x^{16} - 52 x^{15} - 76 x^{14} + 234 x^{13} - 11 x^{12} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[4, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(6802209109663753569286129\) \(\medspace = 11^{16}\cdot 23^{6}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(17.44\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $11^{4/5}23^{1/2}\approx 32.65713384043754$
Ramified primes:   \(11\), \(23\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4939560548782}a^{19}-\frac{915159853377}{4939560548782}a^{18}-\frac{543264947913}{4939560548782}a^{17}-\frac{312986381383}{4939560548782}a^{16}-\frac{546666208669}{2469780274391}a^{15}-\frac{433764354565}{4939560548782}a^{14}+\frac{328543785287}{2469780274391}a^{13}-\frac{586964966431}{4939560548782}a^{12}-\frac{283475911679}{2469780274391}a^{11}+\frac{578899043842}{2469780274391}a^{10}-\frac{647576767959}{4939560548782}a^{9}-\frac{810872401833}{2469780274391}a^{8}+\frac{1234833588683}{4939560548782}a^{7}+\frac{454004797166}{2469780274391}a^{6}+\frac{740363050356}{2469780274391}a^{5}+\frac{874061794987}{4939560548782}a^{4}-\frac{66658474987}{4939560548782}a^{3}+\frac{1025251435322}{2469780274391}a^{2}+\frac{1380124167203}{4939560548782}a+\frac{2156680759065}{4939560548782}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{904112359445}{4939560548782}a^{19}+\frac{275838011071}{4939560548782}a^{18}-\frac{3618926050072}{2469780274391}a^{17}-\frac{1728529844038}{2469780274391}a^{16}+\frac{34821360087461}{4939560548782}a^{15}+\frac{7426896689480}{2469780274391}a^{14}-\frac{134142058231899}{4939560548782}a^{13}+\frac{843569161860}{2469780274391}a^{12}+\frac{166348128571430}{2469780274391}a^{11}-\frac{91284787919534}{2469780274391}a^{10}-\frac{216192853521901}{2469780274391}a^{9}+\frac{465634734514505}{4939560548782}a^{8}+\frac{78967508207329}{2469780274391}a^{7}-\frac{201272954155418}{2469780274391}a^{6}+\frac{149879776051675}{4939560548782}a^{5}-\frac{5796603746839}{2469780274391}a^{4}-\frac{11546091505163}{2469780274391}a^{3}+\frac{67345393896266}{2469780274391}a^{2}-\frac{104915991811385}{4939560548782}a+\frac{9784721446283}{2469780274391}$, $\frac{1911164143085}{2469780274391}a^{19}-\frac{3771711044036}{2469780274391}a^{18}-\frac{19501496373585}{4939560548782}a^{17}+\frac{38149504508285}{4939560548782}a^{16}+\frac{97786762652279}{4939560548782}a^{15}-\frac{99145829004326}{2469780274391}a^{14}-\frac{302283478392645}{4939560548782}a^{13}+\frac{896466546122059}{4939560548782}a^{12}+\frac{3238186328397}{4939560548782}a^{11}-\frac{909327404386225}{2469780274391}a^{10}+\frac{778268204025839}{2469780274391}a^{9}+\frac{924344090186145}{4939560548782}a^{8}-\frac{11\!\cdots\!22}{2469780274391}a^{7}+\frac{11\!\cdots\!09}{4939560548782}a^{6}+\frac{60879876930930}{2469780274391}a^{5}-\frac{656464478477117}{4939560548782}a^{4}+\frac{899596105682261}{4939560548782}a^{3}-\frac{734369061444307}{4939560548782}a^{2}+\frac{281130115530751}{4939560548782}a-\frac{15458294717150}{2469780274391}$, $\frac{6113323554727}{4939560548782}a^{19}-\frac{5867827598720}{2469780274391}a^{18}-\frac{32208203826021}{4939560548782}a^{17}+\frac{59422139222899}{4939560548782}a^{16}+\frac{162146349400477}{4939560548782}a^{15}-\frac{154316967174186}{2469780274391}a^{14}-\frac{512307585329983}{4939560548782}a^{13}+\frac{704239496959037}{2469780274391}a^{12}+\frac{125674911874077}{4939560548782}a^{11}-\frac{29\!\cdots\!47}{4939560548782}a^{10}+\frac{11\!\cdots\!53}{2469780274391}a^{9}+\frac{851053806942616}{2469780274391}a^{8}-\frac{33\!\cdots\!89}{4939560548782}a^{7}+\frac{15\!\cdots\!63}{4939560548782}a^{6}+\frac{157946642413184}{2469780274391}a^{5}-\frac{10\!\cdots\!39}{4939560548782}a^{4}+\frac{13\!\cdots\!59}{4939560548782}a^{3}-\frac{10\!\cdots\!05}{4939560548782}a^{2}+\frac{358237298067771}{4939560548782}a-\frac{34995107214021}{4939560548782}$, $\frac{617960658859}{4939560548782}a^{19}-\frac{238074764357}{2469780274391}a^{18}-\frac{4126591789055}{4939560548782}a^{17}+\frac{1782458620971}{4939560548782}a^{16}+\frac{20280289079153}{4939560548782}a^{15}-\frac{10351872368799}{4939560548782}a^{14}-\frac{36590762623512}{2469780274391}a^{13}+\frac{35657627256835}{2469780274391}a^{12}+\frac{63241309777053}{2469780274391}a^{11}-\frac{213063896160529}{4939560548782}a^{10}-\frac{48361363606725}{4939560548782}a^{9}+\frac{254682189988239}{4939560548782}a^{8}-\frac{102910797909661}{4939560548782}a^{7}-\frac{35862530446447}{2469780274391}a^{6}+\frac{29612371731027}{2469780274391}a^{5}-\frac{52336985634397}{4939560548782}a^{4}+\frac{31933146017832}{2469780274391}a^{3}-\frac{1368355422309}{2469780274391}a^{2}-\frac{18208916136035}{4939560548782}a+\frac{441416369207}{4939560548782}$, $\frac{1653028927565}{4939560548782}a^{19}-\frac{570450083696}{2469780274391}a^{18}-\frac{10622255211005}{4939560548782}a^{17}+\frac{1659062698435}{2469780274391}a^{16}+\frac{51079816749321}{4939560548782}a^{15}-\frac{21659441288491}{4939560548782}a^{14}-\frac{89899709126010}{2469780274391}a^{13}+\frac{83474569632156}{2469780274391}a^{12}+\frac{145315983352872}{2469780274391}a^{11}-\frac{490053155227957}{4939560548782}a^{10}-\frac{35153204245371}{2469780274391}a^{9}+\frac{266937297081767}{2469780274391}a^{8}-\frac{136043155709987}{2469780274391}a^{7}-\frac{39091781826985}{2469780274391}a^{6}+\frac{57041135797349}{2469780274391}a^{5}-\frac{140778160873271}{4939560548782}a^{4}+\frac{81749391804643}{2469780274391}a^{3}-\frac{37088290156399}{4939560548782}a^{2}-\frac{9075401265489}{2469780274391}a+\frac{6715575554273}{4939560548782}$, $\frac{1926995237006}{2469780274391}a^{19}-\frac{4190400544201}{4939560548782}a^{18}-\frac{23132488084887}{4939560548782}a^{17}+\frac{8625631817309}{2469780274391}a^{16}+\frac{112567352809757}{4939560548782}a^{15}-\frac{96433846726187}{4939560548782}a^{14}-\frac{383154029399221}{4939560548782}a^{13}+\frac{272994187485937}{2469780274391}a^{12}+\frac{232899342904268}{2469780274391}a^{11}-\frac{671672132582697}{2469780274391}a^{10}+\frac{373790850974795}{4939560548782}a^{9}+\frac{11\!\cdots\!19}{4939560548782}a^{8}-\frac{11\!\cdots\!61}{4939560548782}a^{7}+\frac{137011623308775}{2469780274391}a^{6}+\frac{86187411307318}{2469780274391}a^{5}-\frac{432661666999775}{4939560548782}a^{4}+\frac{561271833810211}{4939560548782}a^{3}-\frac{149638782685591}{2469780274391}a^{2}+\frac{77948585030293}{4939560548782}a-\frac{9934152307957}{4939560548782}$, $\frac{5457238956001}{4939560548782}a^{19}-\frac{6537122187956}{2469780274391}a^{18}-\frac{25943114627837}{4939560548782}a^{17}+\frac{34341198536093}{2469780274391}a^{16}+\frac{132797548243835}{4939560548782}a^{15}-\frac{351803090504407}{4939560548782}a^{14}-\frac{194961952176347}{2469780274391}a^{13}+\frac{759539385804447}{2469780274391}a^{12}-\frac{133408736384550}{2469780274391}a^{11}-\frac{29\!\cdots\!17}{4939560548782}a^{10}+\frac{14\!\cdots\!86}{2469780274391}a^{9}+\frac{644202377019026}{2469780274391}a^{8}-\frac{19\!\cdots\!58}{2469780274391}a^{7}+\frac{10\!\cdots\!71}{2469780274391}a^{6}+\frac{99705714090469}{2469780274391}a^{5}-\frac{10\!\cdots\!47}{4939560548782}a^{4}+\frac{746832372967378}{2469780274391}a^{3}-\frac{13\!\cdots\!99}{4939560548782}a^{2}+\frac{246573356077692}{2469780274391}a-\frac{48113065071713}{4939560548782}$, $\frac{606677081736}{2469780274391}a^{19}-\frac{1831314822331}{4939560548782}a^{18}-\frac{7421936110977}{4939560548782}a^{17}+\frac{4419098656399}{2469780274391}a^{16}+\frac{37389592778627}{4939560548782}a^{15}-\frac{22992556400535}{2469780274391}a^{14}-\frac{127848156856143}{4939560548782}a^{13}+\frac{229258242643329}{4939560548782}a^{12}+\frac{144596092331467}{4939560548782}a^{11}-\frac{544882859073647}{4939560548782}a^{10}+\frac{91928278281241}{2469780274391}a^{9}+\frac{482168877844439}{4939560548782}a^{8}-\frac{233275946824892}{2469780274391}a^{7}+\frac{44045903186469}{4939560548782}a^{6}+\frac{119411026569779}{4939560548782}a^{5}-\frac{143657216355571}{4939560548782}a^{4}+\frac{101762415501261}{2469780274391}a^{3}-\frac{53932855190715}{2469780274391}a^{2}+\frac{216406342626}{2469780274391}a-\frac{771937794265}{4939560548782}$, $\frac{1251188231817}{4939560548782}a^{19}-\frac{557785099456}{2469780274391}a^{18}-\frac{8002698873307}{4939560548782}a^{17}+\frac{2117470950616}{2469780274391}a^{16}+\frac{19603091052680}{2469780274391}a^{15}-\frac{12147502486752}{2469780274391}a^{14}-\frac{138027493319315}{4939560548782}a^{13}+\frac{77046749318721}{2469780274391}a^{12}+\frac{213049531603541}{4939560548782}a^{11}-\frac{211679142101933}{2469780274391}a^{10}-\frac{7372069391561}{2469780274391}a^{9}+\frac{444013255552101}{4939560548782}a^{8}-\frac{256160820700461}{4939560548782}a^{7}-\frac{58603102622753}{4939560548782}a^{6}+\frac{92191360077149}{4939560548782}a^{5}-\frac{107830937728647}{4939560548782}a^{4}+\frac{70710485663016}{2469780274391}a^{3}-\frac{43971821613579}{4939560548782}a^{2}-\frac{8307283406140}{2469780274391}a+\frac{6560927288505}{4939560548782}$, $\frac{2976214430299}{2469780274391}a^{19}-\frac{3958805537355}{4939560548782}a^{18}-\frac{40855485714653}{4939560548782}a^{17}+\frac{12356623758571}{4939560548782}a^{16}+\frac{100228490893853}{2469780274391}a^{15}-\frac{75011809733987}{4939560548782}a^{14}-\frac{363267978926864}{2469780274391}a^{13}+\frac{300346568833795}{2469780274391}a^{12}+\frac{665022813047505}{2469780274391}a^{11}-\frac{963553339806477}{2469780274391}a^{10}-\frac{372347318018431}{2469780274391}a^{9}+\frac{12\!\cdots\!82}{2469780274391}a^{8}-\frac{373529597695318}{2469780274391}a^{7}-\frac{954606546570773}{4939560548782}a^{6}+\frac{668234933051899}{4939560548782}a^{5}-\frac{416410383859555}{4939560548782}a^{4}+\frac{228946774920951}{2469780274391}a^{3}+\frac{78436578247859}{4939560548782}a^{2}-\frac{125388696823255}{2469780274391}a+\frac{43963461207589}{4939560548782}$, $\frac{988043401857}{4939560548782}a^{19}-\frac{2183368720787}{4939560548782}a^{18}-\frac{4316707093385}{4939560548782}a^{17}+\frac{11270624264763}{4939560548782}a^{16}+\frac{21162482717331}{4939560548782}a^{15}-\frac{59049958760999}{4939560548782}a^{14}-\frac{58331074050623}{4939560548782}a^{13}+\frac{128640513884930}{2469780274391}a^{12}-\frac{82587812549893}{4939560548782}a^{11}-\frac{479546954608071}{4939560548782}a^{10}+\frac{581439058473285}{4939560548782}a^{9}+\frac{119691566152205}{4939560548782}a^{8}-\frac{355137657999212}{2469780274391}a^{7}+\frac{504380329338897}{4939560548782}a^{6}-\frac{7782347940128}{2469780274391}a^{5}-\frac{112734162807180}{2469780274391}a^{4}+\frac{288871004013599}{4939560548782}a^{3}-\frac{263588934820421}{4939560548782}a^{2}+\frac{126216280609713}{4939560548782}a-\frac{14255589977217}{4939560548782}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 26285.5537069 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 26285.5537069 \cdot 1}{2\cdot\sqrt{6802209109663753569286129}}\cr\approx \mathstrut & 0.195848871945 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 5*x^18 + 10*x^17 + 25*x^16 - 52*x^15 - 76*x^14 + 234*x^13 - 11*x^12 - 464*x^11 + 425*x^10 + 206*x^9 - 572*x^8 + 336*x^7 + 3*x^6 - 168*x^5 + 243*x^4 - 204*x^3 + 87*x^2 - 16*x + 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^20 - 2*x^19 - 5*x^18 + 10*x^17 + 25*x^16 - 52*x^15 - 76*x^14 + 234*x^13 - 11*x^12 - 464*x^11 + 425*x^10 + 206*x^9 - 572*x^8 + 336*x^7 + 3*x^6 - 168*x^5 + 243*x^4 - 204*x^3 + 87*x^2 - 16*x + 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^20 - 2*x^19 - 5*x^18 + 10*x^17 + 25*x^16 - 52*x^15 - 76*x^14 + 234*x^13 - 11*x^12 - 464*x^11 + 425*x^10 + 206*x^9 - 572*x^8 + 336*x^7 + 3*x^6 - 168*x^5 + 243*x^4 - 204*x^3 + 87*x^2 - 16*x + 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 - 5*x^18 + 10*x^17 + 25*x^16 - 52*x^15 - 76*x^14 + 234*x^13 - 11*x^12 - 464*x^11 + 425*x^10 + 206*x^9 - 572*x^8 + 336*x^7 + 3*x^6 - 168*x^5 + 243*x^4 - 204*x^3 + 87*x^2 - 16*x + 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2\wr C_5$ (as 20T40):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 160
The 16 conjugacy class representatives for $C_2\wr C_5$
Character table for $C_2\wr C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.2608104505127.1, 10.4.4930254263.1, 10.2.113395848049.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: data not computed
Degree 40 siblings: data not computed
Arithmetically equvalently siblings: data not computed
Minimal sibling: 10.4.4930254263.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.5.0.1}{5} }^{4}$ ${\href{/padicField/3.5.0.1}{5} }^{4}$ ${\href{/padicField/5.10.0.1}{10} }^{2}$ ${\href{/padicField/7.10.0.1}{10} }^{2}$ R ${\href{/padicField/13.5.0.1}{5} }^{4}$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ R ${\href{/padicField/29.5.0.1}{5} }^{4}$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ ${\href{/padicField/37.10.0.1}{10} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.5.0.1}{5} }^{4}$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(11\) Copy content Toggle raw display 11.10.8.5$x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
\(23\) Copy content Toggle raw display $\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$