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Properties

Label 18.18.142...832.1
Degree $18$
Signature $[18, 0]$
Discriminant $1.425\times 10^{26}$
Root discriminant \(28.38\)
Ramified primes $2,37,16361$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_3\wr S_3$ (as 18T319)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 18*x^16 + 43*x^15 + 143*x^14 - 223*x^13 - 603*x^12 + 508*x^11 + 1359*x^10 - 505*x^9 - 1624*x^8 + 146*x^7 + 1012*x^6 + 93*x^5 - 302*x^4 - 70*x^3 + 31*x^2 + 12*x + 1)
 
gp: K = bnfinit(y^18 - 3*y^17 - 18*y^16 + 43*y^15 + 143*y^14 - 223*y^13 - 603*y^12 + 508*y^11 + 1359*y^10 - 505*y^9 - 1624*y^8 + 146*y^7 + 1012*y^6 + 93*y^5 - 302*y^4 - 70*y^3 + 31*y^2 + 12*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 18*x^16 + 43*x^15 + 143*x^14 - 223*x^13 - 603*x^12 + 508*x^11 + 1359*x^10 - 505*x^9 - 1624*x^8 + 146*x^7 + 1012*x^6 + 93*x^5 - 302*x^4 - 70*x^3 + 31*x^2 + 12*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 - 18*x^16 + 43*x^15 + 143*x^14 - 223*x^13 - 603*x^12 + 508*x^11 + 1359*x^10 - 505*x^9 - 1624*x^8 + 146*x^7 + 1012*x^6 + 93*x^5 - 302*x^4 - 70*x^3 + 31*x^2 + 12*x + 1)
 

\( x^{18} - 3 x^{17} - 18 x^{16} + 43 x^{15} + 143 x^{14} - 223 x^{13} - 603 x^{12} + 508 x^{11} + 1359 x^{10} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[18, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(142493532772533353405304832\) \(\medspace = 2^{12}\cdot 37^{9}\cdot 16361^{2}\) 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:  \(28.38\)
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:  $2^{2/3}37^{1/2}16361^{1/2}\approx 1235.0724893715435$
Ramified primes:   \(2\), \(37\), \(16361\) 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:  \(\Q(\sqrt{37}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{486671}a^{17}-\frac{186257}{486671}a^{16}+\frac{229038}{486671}a^{15}-\frac{97104}{486671}a^{14}-\frac{145814}{486671}a^{13}-\frac{234622}{486671}a^{12}+\frac{122953}{486671}a^{11}-\frac{183649}{486671}a^{10}+\frac{177641}{486671}a^{9}-\frac{19384}{486671}a^{8}+\frac{220434}{486671}a^{7}-\frac{175188}{486671}a^{6}+\frac{122898}{486671}a^{5}-\frac{160185}{486671}a^{4}+\frac{217704}{486671}a^{3}+\frac{213492}{486671}a^{2}+\frac{201789}{486671}a+\frac{132923}{486671}$ 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 $1$ (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:  $17$
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:   $\frac{4469269}{486671}a^{17}-\frac{16930945}{486671}a^{16}-\frac{68335174}{486671}a^{15}+\frac{249276116}{486671}a^{14}+\frac{465910112}{486671}a^{13}-\frac{1408133856}{486671}a^{12}-\frac{1773357287}{486671}a^{11}+\frac{3872140118}{486671}a^{10}+\frac{3791199050}{486671}a^{9}-\frac{5589908692}{486671}a^{8}-\frac{4420888680}{486671}a^{7}+\frac{4244591742}{486671}a^{6}+\frac{2706894999}{486671}a^{5}-\frac{1528413891}{486671}a^{4}-\frac{805672221}{486671}a^{3}+\frac{173869754}{486671}a^{2}+\frac{95652999}{486671}a+\frac{7847085}{486671}$, $\frac{1385221}{486671}a^{17}-\frac{6263883}{486671}a^{16}-\frac{17397723}{486671}a^{15}+\frac{92478525}{486671}a^{14}+\frac{90917726}{486671}a^{13}-\frac{537819078}{486671}a^{12}-\frac{265974597}{486671}a^{11}+\frac{1571134363}{486671}a^{10}+\frac{469143472}{486671}a^{9}-\frac{2464526725}{486671}a^{8}-\frac{489337287}{486671}a^{7}+\frac{2053769183}{486671}a^{6}+\frac{286130509}{486671}a^{5}-\frac{830570884}{486671}a^{4}-\frac{93893424}{486671}a^{3}+\frac{123796280}{486671}a^{2}+\frac{16498307}{486671}a-\frac{835170}{486671}$, $\frac{2108220}{486671}a^{17}-\frac{7536255}{486671}a^{16}-\frac{32914022}{486671}a^{15}+\frac{107168877}{486671}a^{14}+\frac{228914795}{486671}a^{13}-\frac{569313666}{486671}a^{12}-\frac{867442095}{486671}a^{11}+\frac{1413371867}{486671}a^{10}+\frac{1764990120}{486671}a^{9}-\frac{1760261617}{486671}a^{8}-\frac{1851526917}{486671}a^{7}+\frac{1115713143}{486671}a^{6}+\frac{959396437}{486671}a^{5}-\frac{324443318}{486671}a^{4}-\frac{220761750}{486671}a^{3}+\frac{26443544}{486671}a^{2}+\frac{17457822}{486671}a+\frac{1871892}{486671}$, $\frac{2929051}{486671}a^{17}-\frac{10227882}{486671}a^{16}-\frac{48977558}{486671}a^{15}+\frac{154405578}{486671}a^{14}+\frac{363043600}{486671}a^{13}-\frac{890565959}{486671}a^{12}-\frac{1469218804}{486671}a^{11}+\frac{2493384734}{486671}a^{10}+\frac{3283684988}{486671}a^{9}-\frac{3667778367}{486671}a^{8}-\frac{3977656281}{486671}a^{7}+\frac{2839360993}{486671}a^{6}+\frac{2523907047}{486671}a^{5}-\frac{1024699210}{486671}a^{4}-\frac{773719117}{486671}a^{3}+\frac{100289037}{486671}a^{2}+\frac{92003662}{486671}a+\frac{9719480}{486671}$, $\frac{374685}{486671}a^{17}-\frac{1516000}{486671}a^{16}-\frac{5867807}{486671}a^{15}+\frac{23958599}{486671}a^{14}+\frac{42194918}{486671}a^{13}-\frac{149909324}{486671}a^{12}-\frac{181647009}{486671}a^{11}+\frac{469524640}{486671}a^{10}+\frac{481176389}{486671}a^{9}-\frac{771189571}{486671}a^{8}-\frac{739848711}{486671}a^{7}+\frac{638921039}{486671}a^{6}+\frac{602699150}{486671}a^{5}-\frac{232844388}{486671}a^{4}-\frac{231185124}{486671}a^{3}+\frac{17604590}{486671}a^{2}+\frac{31685204}{486671}a+\frac{4670838}{486671}$, $\frac{2108220}{486671}a^{17}-\frac{7536255}{486671}a^{16}-\frac{32914022}{486671}a^{15}+\frac{107168877}{486671}a^{14}+\frac{228914795}{486671}a^{13}-\frac{569313666}{486671}a^{12}-\frac{867442095}{486671}a^{11}+\frac{1413371867}{486671}a^{10}+\frac{1764990120}{486671}a^{9}-\frac{1760261617}{486671}a^{8}-\frac{1851526917}{486671}a^{7}+\frac{1115713143}{486671}a^{6}+\frac{959396437}{486671}a^{5}-\frac{324443318}{486671}a^{4}-\frac{220761750}{486671}a^{3}+\frac{26443544}{486671}a^{2}+\frac{17457822}{486671}a+\frac{1385221}{486671}$, $\frac{1847808}{486671}a^{17}-\frac{6583573}{486671}a^{16}-\frac{29786576}{486671}a^{15}+\frac{97057145}{486671}a^{14}+\frac{213038587}{486671}a^{13}-\frac{543273192}{486671}a^{12}-\frac{830806645}{486671}a^{11}+\frac{1461769856}{486671}a^{10}+\frac{1765653933}{486671}a^{9}-\frac{2047809582}{486671}a^{8}-\frac{1981795761}{486671}a^{7}+\frac{1510434569}{486671}a^{6}+\frac{1125695574}{486671}a^{5}-\frac{530727025}{486671}a^{4}-\frac{298085026}{486671}a^{3}+\frac{59980166}{486671}a^{2}+\frac{31108767}{486671}a+\frac{2089162}{486671}$, $\frac{1586280}{486671}a^{17}-\frac{6549938}{486671}a^{16}-\frac{22383224}{486671}a^{15}+\frac{96519264}{486671}a^{14}+\frac{141375524}{486671}a^{13}-\frac{553723889}{486671}a^{12}-\frac{524043116}{486671}a^{11}+\frac{1569605500}{486671}a^{10}+\frac{1176786906}{486671}a^{9}-\frac{2343411934}{486671}a^{8}-\frac{1547761734}{486671}a^{7}+\frac{1813043242}{486671}a^{6}+\frac{1123206928}{486671}a^{5}-\frac{648765078}{486671}a^{4}-\frac{405204068}{486671}a^{3}+\frac{65501588}{486671}a^{2}+\frac{55111952}{486671}a+\frac{5805716}{486671}$, $\frac{5361687}{486671}a^{17}-\frac{17930360}{486671}a^{16}-\frac{90041446}{486671}a^{15}+\frac{261210350}{486671}a^{14}+\frac{669365660}{486671}a^{13}-\frac{1421218968}{486671}a^{12}-\frac{2673558614}{486671}a^{11}+\frac{3628457296}{486671}a^{10}+\frac{5725097318}{486671}a^{9}-\frac{4705537302}{486671}a^{8}-\frac{6411335252}{486671}a^{7}+\frac{3166316288}{486671}a^{6}+\frac{3635674071}{486671}a^{5}-\frac{991773594}{486671}a^{4}-\frac{959935224}{486671}a^{3}+\frac{79162801}{486671}a^{2}+\frac{95838697}{486671}a+\frac{9535359}{486671}$, $\frac{7702916}{486671}a^{17}-\frac{27110845}{486671}a^{16}-\frac{124355328}{486671}a^{15}+\frac{395348328}{486671}a^{14}+\frac{891941929}{486671}a^{13}-\frac{2174805098}{486671}a^{12}-\frac{3480359685}{486671}a^{11}+\frac{5696052834}{486671}a^{10}+\frac{7367542894}{486671}a^{9}-\frac{7697902796}{486671}a^{8}-\frac{8230893317}{486671}a^{7}+\frac{5461356039}{486671}a^{6}+\frac{4708109951}{486671}a^{5}-\frac{1836755873}{486671}a^{4}-\frac{1283571562}{486671}a^{3}+\frac{183954868}{486671}a^{2}+\frac{137051176}{486671}a+\frac{12893460}{486671}$, $\frac{111986}{486671}a^{17}-\frac{430684}{486671}a^{16}-\frac{945587}{486671}a^{15}+\frac{3781648}{486671}a^{14}-\frac{341212}{486671}a^{13}-\frac{472015}{486671}a^{12}+\frac{38565735}{486671}a^{11}-\frac{79214498}{486671}a^{10}-\frac{210100642}{486671}a^{9}+\frac{254344969}{486671}a^{8}+\frac{466339609}{486671}a^{7}-\frac{294357971}{486671}a^{6}-\frac{454751166}{486671}a^{5}+\frac{130156807}{486671}a^{4}+\frac{186898063}{486671}a^{3}-\frac{7384499}{486671}a^{2}-\frac{27305165}{486671}a-\frac{3697496}{486671}$, $\frac{6646612}{486671}a^{17}-\frac{23741835}{486671}a^{16}-\frac{106216533}{486671}a^{15}+\frac{346758213}{486671}a^{14}+\frac{756000741}{486671}a^{13}-\frac{1917539091}{486671}a^{12}-\frac{2947257566}{486671}a^{11}+\frac{5073279966}{486671}a^{10}+\frac{6289485209}{486671}a^{9}-\frac{6952148400}{486671}a^{8}-\frac{7155359748}{486671}a^{7}+\frac{5002429398}{486671}a^{6}+\frac{4209165750}{486671}a^{5}-\frac{1707558414}{486671}a^{4}-\frac{1188009142}{486671}a^{3}+\frac{175375860}{486671}a^{2}+\frac{130901164}{486671}a+\frac{11353710}{486671}$, $\frac{2988404}{486671}a^{17}-\frac{10894509}{486671}a^{16}-\frac{47119503}{486671}a^{15}+\frac{160476571}{486671}a^{14}+\frac{328462010}{486671}a^{13}-\frac{900701622}{486671}a^{12}-\frac{1247785735}{486671}a^{11}+\frac{2446515608}{486671}a^{10}+\frac{2575377741}{486671}a^{9}-\frac{3501345193}{486671}a^{8}-\frac{2788962990}{486671}a^{7}+\frac{2685365508}{486671}a^{6}+\frac{1520531491}{486671}a^{5}-\frac{1000221309}{486671}a^{4}-\frac{385085197}{486671}a^{3}+\frac{125047107}{486671}a^{2}+\frac{39739072}{486671}a+\frac{2374653}{486671}$, $\frac{2415810}{486671}a^{17}-\frac{7416765}{486671}a^{16}-\frac{43342982}{486671}a^{15}+\frac{109355484}{486671}a^{14}+\frac{339086870}{486671}a^{13}-\frac{595018961}{486671}a^{12}-\frac{1394110257}{486671}a^{11}+\frac{1496386339}{486671}a^{10}+\frac{3023129991}{486671}a^{9}-\frac{1885940874}{486671}a^{8}-\frac{3390090279}{486671}a^{7}+\frac{1220916963}{486671}a^{6}+\frac{1902104059}{486671}a^{5}-\frac{351455662}{486671}a^{4}-\frac{486893672}{486671}a^{3}+\frac{13329664}{486671}a^{2}+\frac{44430581}{486671}a+\frac{5346778}{486671}$, $\frac{10976088}{486671}a^{17}-\frac{38686466}{486671}a^{16}-\frac{177782448}{486671}a^{15}+\frac{565605996}{486671}a^{14}+\frac{1285713092}{486671}a^{13}-\frac{3125356623}{486671}a^{12}-\frac{5100427255}{486671}a^{11}+\frac{8231449792}{486671}a^{10}+\frac{11125473029}{486671}a^{9}-\frac{11138534544}{486671}a^{8}-\frac{13054850710}{486671}a^{7}+\frac{7782426123}{486671}a^{6}+\frac{8011250027}{486671}a^{5}-\frac{2482083273}{486671}a^{4}-\frac{2380109095}{486671}a^{3}+\frac{186136077}{486671}a^{2}+\frac{273009404}{486671}a+\frac{29788069}{486671}$, $\frac{6273329}{486671}a^{17}-\frac{21042480}{486671}a^{16}-\frac{104820349}{486671}a^{15}+\frac{305407530}{486671}a^{14}+\frac{776070303}{486671}a^{13}-\frac{1654185872}{486671}a^{12}-\frac{3089634871}{486671}a^{11}+\frac{4199304786}{486671}a^{10}+\frac{6594159615}{486671}a^{9}-\frac{5414861467}{486671}a^{8}-\frac{7368887143}{486671}a^{7}+\frac{3637550506}{486671}a^{6}+\frac{4199253340}{486671}a^{5}-\frac{1144975127}{486671}a^{4}-\frac{1130973945}{486671}a^{3}+\frac{90137120}{486671}a^{2}+\frac{116888443}{486671}a+\frac{11999306}{486671}$, $\frac{1409950}{486671}a^{17}-\frac{4898879}{486671}a^{16}-\frac{22747500}{486671}a^{15}+\frac{70528228}{486671}a^{14}+\frac{162856067}{486671}a^{13}-\frac{380013450}{486671}a^{12}-\frac{629733505}{486671}a^{11}+\frac{964413277}{486671}a^{10}+\frac{1302029396}{486671}a^{9}-\frac{1256585304}{486671}a^{8}-\frac{1391228135}{486671}a^{7}+\frac{877119895}{486671}a^{6}+\frac{740078799}{486671}a^{5}-\frac{308572497}{486671}a^{4}-\frac{178526564}{486671}a^{3}+\frac{37692173}{486671}a^{2}+\frac{16700725}{486671}a+\frac{1188447}{486671}$ Copy content Toggle raw display (assuming GRH)
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:  \( 13397623.9367 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 13397623.9367 \cdot 1}{2\cdot\sqrt{142493532772533353405304832}}\cr\approx \mathstrut & 0.147109301235 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 18*x^16 + 43*x^15 + 143*x^14 - 223*x^13 - 603*x^12 + 508*x^11 + 1359*x^10 - 505*x^9 - 1624*x^8 + 146*x^7 + 1012*x^6 + 93*x^5 - 302*x^4 - 70*x^3 + 31*x^2 + 12*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^18 - 3*x^17 - 18*x^16 + 43*x^15 + 143*x^14 - 223*x^13 - 603*x^12 + 508*x^11 + 1359*x^10 - 505*x^9 - 1624*x^8 + 146*x^7 + 1012*x^6 + 93*x^5 - 302*x^4 - 70*x^3 + 31*x^2 + 12*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^18 - 3*x^17 - 18*x^16 + 43*x^15 + 143*x^14 - 223*x^13 - 603*x^12 + 508*x^11 + 1359*x^10 - 505*x^9 - 1624*x^8 + 146*x^7 + 1012*x^6 + 93*x^5 - 302*x^4 - 70*x^3 + 31*x^2 + 12*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^18 - 3*x^17 - 18*x^16 + 43*x^15 + 143*x^14 - 223*x^13 - 603*x^12 + 508*x^11 + 1359*x^10 - 505*x^9 - 1624*x^8 + 146*x^7 + 1012*x^6 + 93*x^5 - 302*x^4 - 70*x^3 + 31*x^2 + 12*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

$S_3\wr S_3$ (as 18T319):

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 1296
The 22 conjugacy class representatives for $S_3\wr S_3$
Character table for $S_3\wr S_3$

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.148.1 x3, 6.6.810448.1, 9.9.53038958912.1

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 9 sibling: data not computed
Degree 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 27 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 9.9.53038958912.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ ${\href{/padicField/5.6.0.1}{6} }^{3}$ ${\href{/padicField/7.9.0.1}{9} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{6}$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{5}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ R ${\href{/padicField/41.9.0.1}{9} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.3.0.1}{3} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$

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
\(2\) Copy content Toggle raw display 2.18.12.1$x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$$3$$6$$12$$S_3 \times C_3$$[\ ]_{3}^{6}$
\(37\) Copy content Toggle raw display 37.2.1.1$x^{2} + 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} + 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.4.2.1$x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.6.3.1$x^{6} + 2775 x^{5} + 2566998 x^{4} + 791680745 x^{3} + 110476893 x^{2} + 4842384300 x + 27887445532$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(16361\) Copy content Toggle raw display $\Q_{16361}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{16361}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{16361}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{16361}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{16361}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{16361}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{16361}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{16361}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{16361}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{16361}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$