Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 5*x^18 - 14*x^17 + 27*x^16 - 65*x^15 + 127*x^14 - 169*x^13 + 158*x^12 - 226*x^11 - 23*x^10 - 33*x^9 - 106*x^8 - 36*x^7 - 35*x^6 - 27*x^5 - 8*x^4 + 12*x^3 - x^2 + 4*x + 1)
gp: K = bnfinit(y^20 - y^19 + 5*y^18 - 14*y^17 + 27*y^16 - 65*y^15 + 127*y^14 - 169*y^13 + 158*y^12 - 226*y^11 - 23*y^10 - 33*y^9 - 106*y^8 - 36*y^7 - 35*y^6 - 27*y^5 - 8*y^4 + 12*y^3 - y^2 + 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 + 5*x^18 - 14*x^17 + 27*x^16 - 65*x^15 + 127*x^14 - 169*x^13 + 158*x^12 - 226*x^11 - 23*x^10 - 33*x^9 - 106*x^8 - 36*x^7 - 35*x^6 - 27*x^5 - 8*x^4 + 12*x^3 - x^2 + 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 + 5*x^18 - 14*x^17 + 27*x^16 - 65*x^15 + 127*x^14 - 169*x^13 + 158*x^12 - 226*x^11 - 23*x^10 - 33*x^9 - 106*x^8 - 36*x^7 - 35*x^6 - 27*x^5 - 8*x^4 + 12*x^3 - x^2 + 4*x + 1)
\( x^{20} - x^{19} + 5 x^{18} - 14 x^{17} + 27 x^{16} - 65 x^{15} + 127 x^{14} - 169 x^{13} + 158 x^{12} + \cdots + 1 \)
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: |
\(113494176301780230947265625\)
\(\medspace = 5^{10}\cdot 7^{8}\cdot 17^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.08\) | 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: | $5^{1/2}7^{1/2}17^{1/2}\approx 24.392621835300936$ | ||
| Ramified primes: |
\(5\), \(7\), \(17\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{247}a^{16}+\frac{67}{247}a^{14}+\frac{10}{247}a^{13}-\frac{4}{19}a^{12}+\frac{92}{247}a^{11}+\frac{3}{19}a^{10}-\frac{40}{247}a^{9}-\frac{120}{247}a^{8}+\frac{94}{247}a^{7}+\frac{45}{247}a^{6}-\frac{84}{247}a^{5}-\frac{6}{13}a^{4}-\frac{67}{247}a^{3}-\frac{8}{19}a^{2}+\frac{6}{13}a+\frac{73}{247}$, $\frac{1}{247}a^{17}+\frac{67}{247}a^{15}+\frac{10}{247}a^{14}-\frac{4}{19}a^{13}+\frac{92}{247}a^{12}+\frac{3}{19}a^{11}-\frac{40}{247}a^{10}-\frac{120}{247}a^{9}+\frac{94}{247}a^{8}+\frac{45}{247}a^{7}-\frac{84}{247}a^{6}-\frac{6}{13}a^{5}-\frac{67}{247}a^{4}-\frac{8}{19}a^{3}+\frac{6}{13}a^{2}+\frac{73}{247}a$, $\frac{1}{4693}a^{18}+\frac{9}{4693}a^{17}+\frac{5}{4693}a^{16}-\frac{1857}{4693}a^{15}+\frac{2306}{4693}a^{14}+\frac{239}{4693}a^{13}+\frac{386}{4693}a^{12}-\frac{1935}{4693}a^{11}+\frac{1301}{4693}a^{10}+\frac{1741}{4693}a^{9}-\frac{67}{4693}a^{8}-\frac{1802}{4693}a^{7}+\frac{539}{4693}a^{6}-\frac{331}{4693}a^{5}-\frac{2037}{4693}a^{4}+\frac{368}{4693}a^{3}-\frac{357}{4693}a^{2}+\frac{505}{4693}a-\frac{1809}{4693}$, $\frac{1}{35140023829391}a^{19}+\frac{185261880}{1849474938389}a^{18}-\frac{1511628151}{1849474938389}a^{17}+\frac{17511810463}{35140023829391}a^{16}-\frac{26314215296}{142267302953}a^{15}+\frac{577408346901}{2703078756107}a^{14}-\frac{8037665121123}{35140023829391}a^{13}-\frac{1034548587078}{35140023829391}a^{12}+\frac{4722976257897}{35140023829391}a^{11}+\frac{10246092147010}{35140023829391}a^{10}-\frac{2111476404239}{35140023829391}a^{9}-\frac{165774372471}{423373781077}a^{8}-\frac{16966688773845}{35140023829391}a^{7}+\frac{1682105958113}{35140023829391}a^{6}-\frac{8428965278755}{35140023829391}a^{5}+\frac{16543508006229}{35140023829391}a^{4}+\frac{14436616117516}{35140023829391}a^{3}+\frac{4236815101646}{35140023829391}a^{2}-\frac{6497163262333}{35140023829391}a-\frac{11801388893410}{35140023829391}$
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 $1$ (assuming GRH)
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 $2$)
| 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: |
$\frac{1764986242312}{35140023829391}a^{19}-\frac{8477864166962}{35140023829391}a^{18}+\frac{1215666157076}{2703078756107}a^{17}-\frac{56886114817307}{35140023829391}a^{16}+\frac{141096591731715}{35140023829391}a^{15}-\frac{292085648845977}{35140023829391}a^{14}+\frac{644543577052725}{35140023829391}a^{13}-\frac{11\!\cdots\!25}{35140023829391}a^{12}+\frac{13\!\cdots\!49}{35140023829391}a^{11}-\frac{13\!\cdots\!85}{35140023829391}a^{10}+\frac{12\!\cdots\!09}{35140023829391}a^{9}+\frac{2679981358172}{423373781077}a^{8}-\frac{237436748364903}{35140023829391}a^{7}+\frac{543857022737541}{35140023829391}a^{6}+\frac{242581258646611}{35140023829391}a^{5}+\frac{4810485291624}{2703078756107}a^{4}+\frac{24760227081159}{35140023829391}a^{3}+\frac{110864610282950}{35140023829391}a^{2}-\frac{88748333626811}{35140023829391}a+\frac{1013290483455}{35140023829391}$, $\frac{4287820849673}{35140023829391}a^{19}-\frac{260757336943}{2703078756107}a^{18}+\frac{21843721286956}{35140023829391}a^{17}-\frac{57639747458451}{35140023829391}a^{16}+\frac{109693334115436}{35140023829391}a^{15}-\frac{275207807450623}{35140023829391}a^{14}+\frac{527642244654714}{35140023829391}a^{13}-\frac{702247323651829}{35140023829391}a^{12}+\frac{711138531931137}{35140023829391}a^{11}-\frac{10\!\cdots\!31}{35140023829391}a^{10}-\frac{107503909383029}{35140023829391}a^{9}-\frac{4308188888459}{423373781077}a^{8}-\frac{575985215375529}{35140023829391}a^{7}+\frac{636850408471}{2703078756107}a^{6}-\frac{414474053844243}{35140023829391}a^{5}-\frac{87793047895791}{35140023829391}a^{4}+\frac{44717975160688}{35140023829391}a^{3}+\frac{24232485178035}{35140023829391}a^{2}+\frac{37501349174697}{35140023829391}a+\frac{38251603111920}{35140023829391}$, $\frac{8158024946332}{35140023829391}a^{19}-\frac{344784566323}{35140023829391}a^{18}+\frac{33346967802690}{35140023829391}a^{17}-\frac{79035200569966}{35140023829391}a^{16}+\frac{115837187637737}{35140023829391}a^{15}-\frac{341066342296330}{35140023829391}a^{14}+\frac{585307802186226}{35140023829391}a^{13}-\frac{497336553914454}{35140023829391}a^{12}+\frac{227064678190872}{35140023829391}a^{11}-\frac{10\!\cdots\!40}{35140023829391}a^{10}-\frac{13\!\cdots\!26}{35140023829391}a^{9}-\frac{11766057622302}{423373781077}a^{8}-\frac{432757749316736}{35140023829391}a^{7}-\frac{962011807693774}{35140023829391}a^{6}-\frac{529749206790836}{35140023829391}a^{5}-\frac{256288448176834}{35140023829391}a^{4}-\frac{117069307610481}{35140023829391}a^{3}+\frac{119030189629782}{35140023829391}a^{2}+\frac{116823763633026}{35140023829391}a+\frac{45274040674705}{35140023829391}$, $\frac{1039664273751}{35140023829391}a^{19}-\frac{496835752217}{35140023829391}a^{18}+\frac{380853140252}{2703078756107}a^{17}-\frac{11913990806790}{35140023829391}a^{16}+\frac{21832731443563}{35140023829391}a^{15}-\frac{56100954637330}{35140023829391}a^{14}+\frac{102238546061611}{35140023829391}a^{13}-\frac{6399124202191}{1849474938389}a^{12}+\frac{99004188232836}{35140023829391}a^{11}-\frac{179111768345128}{35140023829391}a^{10}-\frac{122384030266833}{35140023829391}a^{9}-\frac{1207555836627}{423373781077}a^{8}-\frac{164270830334514}{35140023829391}a^{7}-\frac{127140236210222}{35140023829391}a^{6}-\frac{5555647366330}{1849474938389}a^{5}-\frac{85615953506315}{35140023829391}a^{4}-\frac{4258822516376}{2703078756107}a^{3}-\frac{17956971340969}{35140023829391}a^{2}+\frac{185966537928}{142267302953}a-\frac{1879827091784}{35140023829391}$, $\frac{6759147991456}{35140023829391}a^{19}-\frac{825207296027}{2703078756107}a^{18}+\frac{35580750472360}{35140023829391}a^{17}-\frac{111090971346228}{35140023829391}a^{16}+\frac{226081427219744}{35140023829391}a^{15}-\frac{510296016654150}{35140023829391}a^{14}+\frac{10\!\cdots\!34}{35140023829391}a^{13}-\frac{77592986869189}{1849474938389}a^{12}+\frac{13\!\cdots\!18}{35140023829391}a^{11}-\frac{126169439842648}{2703078756107}a^{10}+\frac{209319051294767}{35140023829391}a^{9}+\frac{6466682091174}{423373781077}a^{8}-\frac{823041957512486}{35140023829391}a^{7}+\frac{262607168443700}{35140023829391}a^{6}+\frac{2552431127238}{1849474938389}a^{5}+\frac{30031382261170}{35140023829391}a^{4}+\frac{41423311836451}{35140023829391}a^{3}+\frac{119464285870864}{35140023829391}a^{2}-\frac{147073296258}{97340786231}a+\frac{14627005203898}{35140023829391}$, $\frac{1879827091784}{35140023829391}a^{19}-\frac{840162818033}{35140023829391}a^{18}+\frac{684792285131}{2703078756107}a^{17}-\frac{1124552024300}{1849474938389}a^{16}+\frac{38841340671378}{35140023829391}a^{15}-\frac{100356029522397}{35140023829391}a^{14}+\frac{182637086019238}{35140023829391}a^{13}-\frac{215452232449885}{35140023829391}a^{12}+\frac{13494563127711}{2703078756107}a^{11}-\frac{325836734510348}{35140023829391}a^{10}-\frac{222347791456160}{35140023829391}a^{9}-\frac{2221907521635}{423373781077}a^{8}-\frac{299488806169145}{35140023829391}a^{7}-\frac{231944605638738}{35140023829391}a^{6}-\frac{192934184422662}{35140023829391}a^{5}-\frac{156312631438438}{35140023829391}a^{4}-\frac{100654570240587}{35140023829391}a^{3}-\frac{32806767611480}{35140023829391}a^{2}-\frac{19836798432753}{35140023829391}a+\frac{18313019405961}{35140023829391}$, $\frac{10390376303024}{35140023829391}a^{19}-\frac{13201569778984}{35140023829391}a^{18}+\frac{51653512471923}{35140023829391}a^{17}-\frac{157043643579406}{35140023829391}a^{16}+\frac{306140374504149}{35140023829391}a^{15}-\frac{711307937895981}{35140023829391}a^{14}+\frac{14\!\cdots\!40}{35140023829391}a^{13}-\frac{19\!\cdots\!52}{35140023829391}a^{12}+\frac{17\!\cdots\!46}{35140023829391}a^{11}-\frac{24\!\cdots\!39}{35140023829391}a^{10}+\frac{111740823231215}{35140023829391}a^{9}+\frac{2755214984001}{423373781077}a^{8}-\frac{736245136097144}{35140023829391}a^{7}-\frac{131398343986257}{35140023829391}a^{6}-\frac{5501103811135}{2703078756107}a^{5}-\frac{56414349551277}{35140023829391}a^{4}+\frac{106612214373004}{35140023829391}a^{3}+\frac{11582584249753}{1849474938389}a^{2}-\frac{42112727290415}{35140023829391}a-\frac{4054630535922}{35140023829391}$, $\frac{320197098229}{1527827123017}a^{19}-\frac{184153771586}{1527827123017}a^{18}+\frac{1603273375291}{1527827123017}a^{17}-\frac{4070716858865}{1527827123017}a^{16}+\frac{7594460718275}{1527827123017}a^{15}-\frac{1036806857327}{80411953843}a^{14}+\frac{37484274667052}{1527827123017}a^{13}-\frac{49520256154000}{1527827123017}a^{12}+\frac{53712323140354}{1527827123017}a^{11}-\frac{7033485715531}{117525163309}a^{10}+\frac{6431283696441}{1527827123017}a^{9}-\frac{775995898338}{18407555699}a^{8}-\frac{15097801239724}{1527827123017}a^{7}-\frac{1519323385881}{80411953843}a^{6}-\frac{36887009986784}{1527827123017}a^{5}-\frac{9953591909685}{1527827123017}a^{4}-\frac{8999484985635}{1527827123017}a^{3}-\frac{3076869746120}{1527827123017}a^{2}+\frac{1143883152630}{1527827123017}a+\frac{75494561761}{1527827123017}$, $\frac{3037480285631}{35140023829391}a^{19}+\frac{8350335955427}{35140023829391}a^{18}+\frac{5248945983080}{35140023829391}a^{17}+\frac{528463373605}{1849474938389}a^{16}-\frac{68606005261315}{35140023829391}a^{15}+\frac{77825836325681}{35140023829391}a^{14}-\frac{281554829453321}{35140023829391}a^{13}+\frac{784185846267678}{35140023829391}a^{12}-\frac{11\!\cdots\!27}{35140023829391}a^{11}+\frac{606532218352171}{35140023829391}a^{10}-\frac{21\!\cdots\!41}{35140023829391}a^{9}-\frac{9825810339160}{423373781077}a^{8}-\frac{381085735551751}{35140023829391}a^{7}-\frac{950362967412398}{35140023829391}a^{6}-\frac{703308637228178}{35140023829391}a^{5}-\frac{391674034861495}{35140023829391}a^{4}-\frac{204360443261708}{35140023829391}a^{3}-\frac{7064096818185}{35140023829391}a^{2}+\frac{106094327093093}{35140023829391}a+\frac{3604433279073}{35140023829391}$, $\frac{258321523287}{1849474938389}a^{19}+\frac{723400573091}{35140023829391}a^{18}+\frac{17606695646573}{35140023829391}a^{17}-\frac{41617011279850}{35140023829391}a^{16}+\frac{48538050876864}{35140023829391}a^{15}-\frac{158797722737509}{35140023829391}a^{14}+\frac{250116337612811}{35140023829391}a^{13}-\frac{76431133327834}{35140023829391}a^{12}-\frac{222608462027077}{35140023829391}a^{11}-\frac{17197757285459}{2703078756107}a^{10}-\frac{13\!\cdots\!15}{35140023829391}a^{9}-\frac{1767564667121}{423373781077}a^{8}-\frac{346472251757880}{35140023829391}a^{7}-\frac{437987097385687}{35140023829391}a^{6}-\frac{210006629915011}{35140023829391}a^{5}-\frac{164465970051154}{35140023829391}a^{4}+\frac{29244787530651}{35140023829391}a^{3}+\frac{80685335488822}{35140023829391}a^{2}+\frac{140268485376049}{35140023829391}a+\frac{41036311582614}{35140023829391}$, $\frac{23481441161}{1527827123017}a^{19}-\frac{271172296275}{1527827123017}a^{18}+\frac{465837395433}{1527827123017}a^{17}-\frac{1687202918385}{1527827123017}a^{16}+\frac{4651570279416}{1527827123017}a^{15}-\frac{9734368082557}{1527827123017}a^{14}+\frac{22200429534175}{1527827123017}a^{13}-\frac{42802999824072}{1527827123017}a^{12}+\frac{60170842916321}{1527827123017}a^{11}-\frac{65357168942920}{1527827123017}a^{10}+\frac{77165803285656}{1527827123017}a^{9}-\frac{287039868723}{18407555699}a^{8}+\frac{9745928452925}{1527827123017}a^{7}+\frac{19308893890461}{1527827123017}a^{6}-\frac{7546889804663}{1527827123017}a^{5}+\frac{6939405642808}{1527827123017}a^{4}-\frac{3689138434187}{1527827123017}a^{3}-\frac{78785557729}{1527827123017}a^{2}-\frac{3864096554686}{1527827123017}a+\frac{597592704199}{1527827123017}$
| 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: | \( 106567.833621 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 106567.833621 \cdot 1}{2\cdot\sqrt{113494176301780230947265625}}\cr\approx \mathstrut & 0.194387476771 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 5*x^18 - 14*x^17 + 27*x^16 - 65*x^15 + 127*x^14 - 169*x^13 + 158*x^12 - 226*x^11 - 23*x^10 - 33*x^9 - 106*x^8 - 36*x^7 - 35*x^6 - 27*x^5 - 8*x^4 + 12*x^3 - x^2 + 4*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 - x^19 + 5*x^18 - 14*x^17 + 27*x^16 - 65*x^15 + 127*x^14 - 169*x^13 + 158*x^12 - 226*x^11 - 23*x^10 - 33*x^9 - 106*x^8 - 36*x^7 - 35*x^6 - 27*x^5 - 8*x^4 + 12*x^3 - x^2 + 4*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 - x^19 + 5*x^18 - 14*x^17 + 27*x^16 - 65*x^15 + 127*x^14 - 169*x^13 + 158*x^12 - 226*x^11 - 23*x^10 - 33*x^9 - 106*x^8 - 36*x^7 - 35*x^6 - 27*x^5 - 8*x^4 + 12*x^3 - x^2 + 4*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 - x^19 + 5*x^18 - 14*x^17 + 27*x^16 - 65*x^15 + 127*x^14 - 169*x^13 + 158*x^12 - 226*x^11 - 23*x^10 - 33*x^9 - 106*x^8 - 36*x^7 - 35*x^6 - 27*x^5 - 8*x^4 + 12*x^3 - x^2 + 4*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
$C_2\times D_{10}$ (as 20T8):
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 40 |
| The 16 conjugacy class representatives for $C_2\times D_{10}$ |
| Character table for $C_2\times D_{10}$ |
Intermediate fields
| \(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}, \sqrt{17})\), 5.1.14161.1, 10.2.626668503125.1, 10.2.10653364553125.1, 10.2.3409076657.1 |
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
| Galois closure: | deg 40 |
| Degree 20 siblings: | 20.0.19242957227637478603515625.1, deg 20, deg 20 |
| Minimal sibling: | 20.0.19242957227637478603515625.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | R | R | ${\href{/padicField/11.2.0.1}{2} }^{10}$ | ${\href{/padicField/13.2.0.1}{2} }^{10}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{8}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.2.0.1}{2} }^{10}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{10}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{10}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# 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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $20$ | $2$ | $10$ | $10$ | |||
|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(17\)
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |