Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 + 8*x^19 - 12*x^18 + 12*x^17 - 16*x^16 + 44*x^15 - 94*x^14 + 160*x^13 - 200*x^12 + 184*x^11 - 132*x^10 + 52*x^9 - 8*x^7 + 4*x^6 + 16*x^4 - 36*x^3 + 32*x^2 - 16*x + 4)
gp: K = bnfinit(y^21 - 4*y^20 + 8*y^19 - 12*y^18 + 12*y^17 - 16*y^16 + 44*y^15 - 94*y^14 + 160*y^13 - 200*y^12 + 184*y^11 - 132*y^10 + 52*y^9 - 8*y^7 + 4*y^6 + 16*y^4 - 36*y^3 + 32*y^2 - 16*y + 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 4*x^20 + 8*x^19 - 12*x^18 + 12*x^17 - 16*x^16 + 44*x^15 - 94*x^14 + 160*x^13 - 200*x^12 + 184*x^11 - 132*x^10 + 52*x^9 - 8*x^7 + 4*x^6 + 16*x^4 - 36*x^3 + 32*x^2 - 16*x + 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 4*x^20 + 8*x^19 - 12*x^18 + 12*x^17 - 16*x^16 + 44*x^15 - 94*x^14 + 160*x^13 - 200*x^12 + 184*x^11 - 132*x^10 + 52*x^9 - 8*x^7 + 4*x^6 + 16*x^4 - 36*x^3 + 32*x^2 - 16*x + 4)
\( x^{21} - 4 x^{20} + 8 x^{19} - 12 x^{18} + 12 x^{17} - 16 x^{16} + 44 x^{15} - 94 x^{14} + 160 x^{13} + \cdots + 4 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[3, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-20735011601517857380131405824\)
\(\medspace = -\,2^{20}\cdot 11^{7}\cdot 317^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.31\) | 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: | $2^{20/21}11^{1/2}317^{1/2}\approx 114.26710056592353$ | ||
| Ramified primes: |
\(2\), \(11\), \(317\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{2}a^{16}$, $\frac{1}{2}a^{17}$, $\frac{1}{2}a^{18}$, $\frac{1}{2}a^{19}$, $\frac{1}{184199525122}a^{20}-\frac{19449544574}{92099762561}a^{19}-\frac{8681013010}{92099762561}a^{18}-\frac{7731714081}{184199525122}a^{17}-\frac{13637691330}{92099762561}a^{16}+\frac{5944679667}{92099762561}a^{15}-\frac{23971240947}{184199525122}a^{14}+\frac{3866440491}{92099762561}a^{13}+\frac{4616174469}{92099762561}a^{12}+\frac{6256310464}{92099762561}a^{11}-\frac{40232710432}{92099762561}a^{10}+\frac{31275334049}{92099762561}a^{9}-\frac{45218409066}{92099762561}a^{8}+\frac{25890848273}{92099762561}a^{7}-\frac{45971900466}{92099762561}a^{6}+\frac{20590808210}{92099762561}a^{5}-\frac{11220897325}{92099762561}a^{4}+\frac{10753877908}{92099762561}a^{3}+\frac{18226363094}{92099762561}a^{2}-\frac{23402472648}{92099762561}a-\frac{883829038}{92099762561}$
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: |
$\frac{115682488833}{184199525122}a^{20}-\frac{167680603401}{92099762561}a^{19}+\frac{276182824408}{92099762561}a^{18}-\frac{399929726013}{92099762561}a^{17}+\frac{283344143778}{92099762561}a^{16}-\frac{647826004742}{92099762561}a^{15}+\frac{3763359732679}{184199525122}a^{14}-\frac{3358853340608}{92099762561}a^{13}+\frac{5608659088822}{92099762561}a^{12}-\frac{5706783431448}{92099762561}a^{11}+\frac{4801758210004}{92099762561}a^{10}-\frac{3076517830208}{92099762561}a^{9}+\frac{129572117790}{92099762561}a^{8}-\frac{15466631874}{92099762561}a^{7}-\frac{516718983102}{92099762561}a^{6}+\frac{145224817106}{92099762561}a^{5}+\frac{12524009836}{92099762561}a^{4}+\frac{975320597064}{92099762561}a^{3}-\frac{1069325898224}{92099762561}a^{2}+\frac{683612304696}{92099762561}a-\frac{364520838409}{92099762561}$, $\frac{126662348989}{92099762561}a^{20}-\frac{412140714893}{92099762561}a^{19}+\frac{1409726591669}{184199525122}a^{18}-\frac{1973071337349}{184199525122}a^{17}+\frac{771951777111}{92099762561}a^{16}-\frac{1445509849090}{92099762561}a^{15}+\frac{4494955552146}{92099762561}a^{14}-\frac{8559848136588}{92099762561}a^{13}+\frac{13822746547260}{92099762561}a^{12}-\frac{14865366463190}{92099762561}a^{11}+\frac{12098751984944}{92099762561}a^{10}-\frac{7626196942979}{92099762561}a^{9}+\frac{984551077936}{92099762561}a^{8}+\frac{562871293524}{92099762561}a^{7}-\frac{462147151916}{92099762561}a^{6}-\frac{69673604512}{92099762561}a^{5}+\frac{26205969738}{92099762561}a^{4}+\frac{2069964962015}{92099762561}a^{3}-\frac{2926427762480}{92099762561}a^{2}+\frac{1792180474108}{92099762561}a-\frac{625140777407}{92099762561}$, $\frac{108749721201}{92099762561}a^{20}-\frac{355929369905}{92099762561}a^{19}+\frac{1212113761249}{184199525122}a^{18}-\frac{1689357266973}{184199525122}a^{17}+\frac{655131787164}{92099762561}a^{16}-\frac{2438261871017}{184199525122}a^{15}+\frac{3865179344592}{92099762561}a^{14}-\frac{7358174434277}{92099762561}a^{13}+\frac{23700319468503}{184199525122}a^{12}-\frac{12700068761811}{92099762561}a^{11}+\frac{10126227974596}{92099762561}a^{10}-\frac{6320149985254}{92099762561}a^{9}+\frac{568664721559}{92099762561}a^{8}+\frac{653224247024}{92099762561}a^{7}-\frac{435624434465}{92099762561}a^{6}+\frac{68372937928}{92099762561}a^{5}-\frac{22721508988}{92099762561}a^{4}+\frac{1845078461087}{92099762561}a^{3}-\frac{2546850515646}{92099762561}a^{2}+\frac{1503514667710}{92099762561}a-\frac{483054532119}{92099762561}$, $\frac{233182354285}{92099762561}a^{20}-\frac{760948113755}{92099762561}a^{19}+\frac{1297411045876}{92099762561}a^{18}-\frac{3613321222379}{184199525122}a^{17}+\frac{1398946750690}{92099762561}a^{16}-\frac{2616849588176}{92099762561}a^{15}+\frac{8258390785024}{92099762561}a^{14}-\frac{15751753767589}{92099762561}a^{13}+\frac{50714162383693}{184199525122}a^{12}-\frac{27127837218672}{92099762561}a^{11}+\frac{21713027218998}{92099762561}a^{10}-\frac{13409762463422}{92099762561}a^{9}+\frac{1248760337702}{92099762561}a^{8}+\frac{1396167315296}{92099762561}a^{7}-\frac{1015344059509}{92099762561}a^{6}+\frac{103540618640}{92099762561}a^{5}-\frac{151013810296}{92099762561}a^{4}+\frac{3974279950433}{92099762561}a^{3}-\frac{5445839064782}{92099762561}a^{2}+\frac{3268569688164}{92099762561}a-\frac{951322495615}{92099762561}$, $\frac{11686259697}{184199525122}a^{20}-\frac{18866527523}{92099762561}a^{19}+\frac{17982382577}{92099762561}a^{18}-\frac{10821063207}{92099762561}a^{17}-\frac{5710237856}{92099762561}a^{16}-\frac{31017766489}{184199525122}a^{15}+\frac{393087536407}{184199525122}a^{14}-\frac{287086677210}{92099762561}a^{13}+\frac{252806363030}{92099762561}a^{12}-\frac{134247331184}{92099762561}a^{11}-\frac{243231777234}{92099762561}a^{10}+\frac{198345238340}{92099762561}a^{9}-\frac{145038749664}{92099762561}a^{8}-\frac{38522395896}{92099762561}a^{7}+\frac{430578822542}{92099762561}a^{6}-\frac{132700807270}{92099762561}a^{5}+\frac{37336676564}{92099762561}a^{4}+\frac{37638303706}{92099762561}a^{3}-\frac{97981618408}{92099762561}a^{2}-\frac{122673232441}{92099762561}a+\frac{133155860743}{92099762561}$, $\frac{64394675621}{184199525122}a^{20}-\frac{98007684128}{92099762561}a^{19}+\frac{164989955423}{92099762561}a^{18}-\frac{228108697024}{92099762561}a^{17}+\frac{163044618701}{92099762561}a^{16}-\frac{699858067835}{184199525122}a^{15}+\frac{1060944515553}{92099762561}a^{14}-\frac{2007686447239}{92099762561}a^{13}+\frac{3231041861833}{92099762561}a^{12}-\frac{6585923491487}{184199525122}a^{11}+\frac{2674948683566}{92099762561}a^{10}-\frac{1431283123170}{92099762561}a^{9}+\frac{27695615399}{92099762561}a^{8}+\frac{207972943496}{92099762561}a^{7}-\frac{251389607072}{92099762561}a^{6}-\frac{84215406653}{92099762561}a^{5}-\frac{155479952452}{92099762561}a^{4}+\frac{441810188458}{92099762561}a^{3}-\frac{696089455224}{92099762561}a^{2}+\frac{366343825061}{92099762561}a-\frac{126515291523}{92099762561}$, $\frac{80366310819}{92099762561}a^{20}-\frac{259113241244}{92099762561}a^{19}+\frac{425495807665}{92099762561}a^{18}-\frac{1167193646577}{184199525122}a^{17}+\frac{433836395633}{92099762561}a^{16}-\frac{1701498212613}{184199525122}a^{15}+\frac{2814619600626}{92099762561}a^{14}-\frac{5214307684934}{92099762561}a^{13}+\frac{16525863442107}{184199525122}a^{12}-\frac{8681020773285}{92099762561}a^{11}+\frac{6593956244262}{92099762561}a^{10}-\frac{4113710645602}{92099762561}a^{9}+\frac{12631759635}{92099762561}a^{8}+\frac{581296354320}{92099762561}a^{7}-\frac{120048523586}{92099762561}a^{6}+\frac{183413378116}{92099762561}a^{5}+\frac{150289704678}{92099762561}a^{4}+\frac{1337580954127}{92099762561}a^{3}-\frac{1741055579878}{92099762561}a^{2}+\frac{829256477570}{92099762561}a-\frac{318883904377}{92099762561}$, $\frac{51919628741}{184199525122}a^{20}-\frac{245889580095}{184199525122}a^{19}+\frac{508231927795}{184199525122}a^{18}-\frac{369293482188}{92099762561}a^{17}+\frac{752304269049}{184199525122}a^{16}-\frac{833405553291}{184199525122}a^{15}+\frac{2562650206357}{184199525122}a^{14}-\frac{5891706315395}{184199525122}a^{13}+\frac{9839903116083}{184199525122}a^{12}-\frac{6260705598458}{92099762561}a^{11}+\frac{5374938244893}{92099762561}a^{10}-\frac{3619458793839}{92099762561}a^{9}+\frac{1172428946813}{92099762561}a^{8}+\frac{588297593380}{92099762561}a^{7}-\frac{326594196378}{92099762561}a^{6}+\frac{126729335009}{92099762561}a^{5}+\frac{70880406917}{92099762561}a^{4}+\frac{558819632233}{92099762561}a^{3}-\frac{1207853245422}{92099762561}a^{2}+\frac{1002403813448}{92099762561}a-\frac{355528811935}{92099762561}$, $\frac{64120729340}{92099762561}a^{20}-\frac{214720420484}{92099762561}a^{19}+\frac{772557536883}{184199525122}a^{18}-\frac{558442590497}{92099762561}a^{17}+\frac{943317162319}{184199525122}a^{16}-\frac{813549256047}{92099762561}a^{15}+\frac{2366782070401}{92099762561}a^{14}-\frac{4639859194467}{92099762561}a^{13}+\frac{7700067306556}{92099762561}a^{12}-\frac{17251010292185}{184199525122}a^{11}+\frac{7502252305124}{92099762561}a^{10}-\frac{5011732551490}{92099762561}a^{9}+\frac{1287198002770}{92099762561}a^{8}-\frac{69364732772}{92099762561}a^{7}-\frac{346970892871}{92099762561}a^{6}-\frac{36408241162}{92099762561}a^{5}-\frac{56891931962}{92099762561}a^{4}+\frac{1113300707200}{92099762561}a^{3}-\frac{1641959216011}{92099762561}a^{2}+\frac{1177264220541}{92099762561}a-\frac{410120431905}{92099762561}$, $\frac{3157422423}{92099762561}a^{20}-\frac{131478780441}{184199525122}a^{19}+\frac{166255478320}{92099762561}a^{18}-\frac{244091126104}{92099762561}a^{17}+\frac{318460454136}{92099762561}a^{16}-\frac{195231227951}{92099762561}a^{15}+\frac{1284064396437}{184199525122}a^{14}-\frac{3759543926401}{184199525122}a^{13}+\frac{3109130087891}{92099762561}a^{12}-\frac{9522816809895}{184199525122}a^{11}+\frac{4203980853762}{92099762561}a^{10}-\frac{3076654728221}{92099762561}a^{9}+\frac{1631047953050}{92099762561}a^{8}+\frac{519204699789}{92099762561}a^{7}+\frac{128664878853}{92099762561}a^{6}+\frac{52661440190}{92099762561}a^{5}+\frac{149450843997}{92099762561}a^{4}+\frac{16316243814}{92099762561}a^{3}-\frac{922987022585}{92099762561}a^{2}+\frac{777560920275}{92099762561}a-\frac{305055952417}{92099762561}$, $\frac{287905177321}{184199525122}a^{20}-\frac{444102101498}{92099762561}a^{19}+\frac{728171963084}{92099762561}a^{18}-\frac{1013135891135}{92099762561}a^{17}+\frac{736320889009}{92099762561}a^{16}-\frac{1549160001401}{92099762561}a^{15}+\frac{4870848103723}{92099762561}a^{14}-\frac{8922369001937}{92099762561}a^{13}+\frac{28667250233153}{184199525122}a^{12}-\frac{29660150478249}{184199525122}a^{11}+\frac{11712467903898}{92099762561}a^{10}-\frac{7169719171842}{92099762561}a^{9}+\frac{175793916341}{92099762561}a^{8}+\frac{588754882788}{92099762561}a^{7}-\frac{470288641354}{92099762561}a^{6}+\frac{33406780252}{92099762561}a^{5}+\frac{12439580280}{92099762561}a^{4}+\frac{2313794027117}{92099762561}a^{3}-\frac{3015080038148}{92099762561}a^{2}+\frac{1791358375207}{92099762561}a-\frac{627183638729}{92099762561}$
| 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: | \( 999740.077738 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 999740.077738 \cdot 1}{2\cdot\sqrt{20735011601517857380131405824}}\cr\approx \mathstrut & 0.423851542233 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 + 8*x^19 - 12*x^18 + 12*x^17 - 16*x^16 + 44*x^15 - 94*x^14 + 160*x^13 - 200*x^12 + 184*x^11 - 132*x^10 + 52*x^9 - 8*x^7 + 4*x^6 + 16*x^4 - 36*x^3 + 32*x^2 - 16*x + 4)
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^21 - 4*x^20 + 8*x^19 - 12*x^18 + 12*x^17 - 16*x^16 + 44*x^15 - 94*x^14 + 160*x^13 - 200*x^12 + 184*x^11 - 132*x^10 + 52*x^9 - 8*x^7 + 4*x^6 + 16*x^4 - 36*x^3 + 32*x^2 - 16*x + 4, 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^21 - 4*x^20 + 8*x^19 - 12*x^18 + 12*x^17 - 16*x^16 + 44*x^15 - 94*x^14 + 160*x^13 - 200*x^12 + 184*x^11 - 132*x^10 + 52*x^9 - 8*x^7 + 4*x^6 + 16*x^4 - 36*x^3 + 32*x^2 - 16*x + 4);
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^21 - 4*x^20 + 8*x^19 - 12*x^18 + 12*x^17 - 16*x^16 + 44*x^15 - 94*x^14 + 160*x^13 - 200*x^12 + 184*x^11 - 132*x^10 + 52*x^9 - 8*x^7 + 4*x^6 + 16*x^4 - 36*x^3 + 32*x^2 - 16*x + 4);
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
$S_3\times \GL(3,2)$ (as 21T27):
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 non-solvable group of order 1008 |
| The 18 conjugacy class representatives for $S_3\times \GL(3,2)$ |
| Character table for $S_3\times \GL(3,2)$ |
Intermediate fields
| 3.1.44.1, 7.3.6431296.2 |
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 24 sibling: | data not computed |
| Degree 42 siblings: | data not computed |
| Arithmetically equvalently siblings: | data not computed |
| Minimal sibling: | 21.3.20735011601517857380131405824.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $21$ | $21$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.7.0.1}{7} }$ | R | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $21$ | $21$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.7.0.1}{7} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.21.20.1 | $x^{21} + 2$ | $21$ | $1$ | $20$ | 21T11 | $[\ ]_{21}^{6}$ |
|
\(11\)
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(317\)
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $12$ | $2$ | $6$ | $6$ |