↑

Properties

Label 18.0.272...616.4
Degree $18$
Signature $[0, 9]$
Discriminant $-2.728\times 10^{24}$
Root discriminant \(22.78\)
Ramified primes $2,3,11$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 - 12*x^15 + 18*x^14 - 24*x^13 + 155*x^12 - 198*x^11 + 198*x^10 - 382*x^9 + 1305*x^8 - 1326*x^7 + 1667*x^6 - 2772*x^5 + 5418*x^4 - 4584*x^3 + 3528*x^2 - 2340*x + 676)
 
gp: K = bnfinit(y^18 + 3*y^16 - 12*y^15 + 18*y^14 - 24*y^13 + 155*y^12 - 198*y^11 + 198*y^10 - 382*y^9 + 1305*y^8 - 1326*y^7 + 1667*y^6 - 2772*y^5 + 5418*y^4 - 4584*y^3 + 3528*y^2 - 2340*y + 676, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 3*x^16 - 12*x^15 + 18*x^14 - 24*x^13 + 155*x^12 - 198*x^11 + 198*x^10 - 382*x^9 + 1305*x^8 - 1326*x^7 + 1667*x^6 - 2772*x^5 + 5418*x^4 - 4584*x^3 + 3528*x^2 - 2340*x + 676);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 3*x^16 - 12*x^15 + 18*x^14 - 24*x^13 + 155*x^12 - 198*x^11 + 198*x^10 - 382*x^9 + 1305*x^8 - 1326*x^7 + 1667*x^6 - 2772*x^5 + 5418*x^4 - 4584*x^3 + 3528*x^2 - 2340*x + 676)
 

\( x^{18} + 3 x^{16} - 12 x^{15} + 18 x^{14} - 24 x^{13} + 155 x^{12} - 198 x^{11} + 198 x^{10} + \cdots + 676 \) 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:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-2727747884710191986159616\) \(\medspace = -\,2^{12}\cdot 3^{24}\cdot 11^{9}\) 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:  \(22.78\)
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:  $2^{2/3}3^{4/3}11^{1/2}\approx 22.779525808351707$
Ramified primes:   \(2\), \(3\), \(11\) 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:  \(\Q(\sqrt{-11}) \)
$\card{ \Gal(K/\Q) }$:  $18$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{22}a^{12}-\frac{2}{11}a^{11}-\frac{2}{11}a^{10}-\frac{5}{22}a^{9}-\frac{2}{11}a^{8}+\frac{1}{11}a^{7}+\frac{5}{22}a^{6}+\frac{4}{11}a^{5}-\frac{4}{11}a^{4}+\frac{7}{22}a^{3}-\frac{1}{11}a^{2}-\frac{4}{11}a-\frac{5}{11}$, $\frac{1}{22}a^{13}+\frac{1}{11}a^{11}+\frac{1}{22}a^{10}-\frac{1}{11}a^{9}+\frac{4}{11}a^{8}-\frac{9}{22}a^{7}+\frac{3}{11}a^{6}+\frac{1}{11}a^{5}-\frac{3}{22}a^{4}+\frac{2}{11}a^{3}+\frac{3}{11}a^{2}+\frac{1}{11}a+\frac{2}{11}$, $\frac{1}{22}a^{14}-\frac{1}{11}a^{11}-\frac{5}{22}a^{10}-\frac{2}{11}a^{9}+\frac{5}{11}a^{8}+\frac{1}{11}a^{7}+\frac{3}{22}a^{6}-\frac{4}{11}a^{5}-\frac{1}{11}a^{4}-\frac{4}{11}a^{3}+\frac{3}{11}a^{2}-\frac{1}{11}a-\frac{1}{11}$, $\frac{1}{44044}a^{15}-\frac{255}{44044}a^{14}+\frac{101}{6292}a^{13}-\frac{97}{11011}a^{12}-\frac{4897}{44044}a^{11}+\frac{8725}{44044}a^{10}+\frac{6001}{44044}a^{9}+\frac{3525}{22022}a^{8}+\frac{13775}{44044}a^{7}-\frac{18855}{44044}a^{6}-\frac{707}{6292}a^{5}-\frac{444}{11011}a^{4}+\frac{2831}{11011}a^{3}+\frac{10519}{22022}a^{2}-\frac{4649}{11011}a+\frac{281}{847}$, $\frac{1}{1893892}a^{16}-\frac{1}{172172}a^{15}+\frac{32581}{1893892}a^{14}-\frac{463}{36421}a^{13}+\frac{16547}{1893892}a^{12}-\frac{7287}{270556}a^{11}+\frac{277045}{1893892}a^{10}-\frac{1915}{43043}a^{9}-\frac{151909}{1893892}a^{8}-\frac{197291}{1893892}a^{7}+\frac{629661}{1893892}a^{6}-\frac{107169}{946946}a^{5}+\frac{152353}{946946}a^{4}-\frac{195367}{473473}a^{3}+\frac{31373}{67639}a^{2}+\frac{4780}{67639}a-\frac{6357}{36421}$, $\frac{1}{66\!\cdots\!76}a^{17}-\frac{168617940}{15\!\cdots\!29}a^{16}+\frac{13815551780}{12\!\cdots\!63}a^{15}-\frac{14\!\cdots\!53}{66\!\cdots\!76}a^{14}+\frac{14822403372625}{869065388007388}a^{13}+\frac{285322963437}{116581942293674}a^{12}+\frac{22\!\cdots\!57}{16\!\cdots\!19}a^{11}+\frac{238614552788281}{51\!\cdots\!52}a^{10}+\frac{15\!\cdots\!41}{66\!\cdots\!76}a^{9}-\frac{14\!\cdots\!53}{33\!\cdots\!38}a^{8}+\frac{29\!\cdots\!81}{16\!\cdots\!19}a^{7}-\frac{18\!\cdots\!57}{66\!\cdots\!76}a^{6}-\frac{77350181361316}{183840755155409}a^{5}-\frac{55\!\cdots\!53}{16\!\cdots\!19}a^{4}-\frac{22\!\cdots\!69}{16\!\cdots\!19}a^{3}-\frac{328848656218828}{16\!\cdots\!19}a^{2}+\frac{124777276995494}{389058342305633}a-\frac{69791010415649}{12\!\cdots\!63}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{3}$, which has order $3$ (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:  $8$
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:   $\frac{3673662946209}{16\!\cdots\!19}a^{17}+\frac{24493832439}{70737880419206}a^{16}+\frac{6973702308389}{51\!\cdots\!52}a^{15}-\frac{532853009253}{467958285850132}a^{14}+\frac{124204524845007}{66\!\cdots\!76}a^{13}-\frac{4810004633531}{816073596055718}a^{12}+\frac{314944610126697}{95\!\cdots\!68}a^{11}+\frac{8080763527971}{95\!\cdots\!68}a^{10}+\frac{45\!\cdots\!73}{66\!\cdots\!76}a^{9}-\frac{15\!\cdots\!97}{16\!\cdots\!19}a^{8}+\frac{14\!\cdots\!43}{66\!\cdots\!76}a^{7}+\frac{380385308253677}{66\!\cdots\!76}a^{6}+\frac{47\!\cdots\!95}{66\!\cdots\!76}a^{5}-\frac{84\!\cdots\!77}{16\!\cdots\!19}a^{4}+\frac{12\!\cdots\!10}{16\!\cdots\!19}a^{3}-\frac{11\!\cdots\!65}{33\!\cdots\!38}a^{2}+\frac{33\!\cdots\!67}{16\!\cdots\!19}a-\frac{192645374744407}{12\!\cdots\!63}$, $\frac{37232929094859}{33\!\cdots\!38}a^{17}+\frac{446106619809}{467958285850132}a^{16}+\frac{125669299788605}{33\!\cdots\!38}a^{15}-\frac{182957124271707}{16\!\cdots\!19}a^{14}+\frac{624674785676313}{66\!\cdots\!76}a^{13}-\frac{26049215721517}{16\!\cdots\!36}a^{12}+\frac{492512336117631}{30\!\cdots\!58}a^{11}-\frac{27\!\cdots\!31}{33\!\cdots\!38}a^{10}+\frac{71\!\cdots\!57}{66\!\cdots\!76}a^{9}-\frac{22\!\cdots\!97}{60\!\cdots\!16}a^{8}+\frac{39\!\cdots\!79}{33\!\cdots\!38}a^{7}-\frac{67\!\cdots\!60}{16\!\cdots\!19}a^{6}+\frac{81\!\cdots\!37}{60\!\cdots\!16}a^{5}-\frac{74\!\cdots\!45}{33\!\cdots\!38}a^{4}+\frac{71\!\cdots\!41}{16\!\cdots\!19}a^{3}-\frac{46\!\cdots\!83}{30\!\cdots\!58}a^{2}+\frac{632095067921430}{217266347001847}a-\frac{43186350229334}{29927564792741}$, $\frac{384569431293}{778116684611266}a^{17}+\frac{70058777941}{141475760838412}a^{16}+\frac{480542963149}{389058342305633}a^{15}-\frac{1913486862503}{389058342305633}a^{14}+\frac{3995531185953}{15\!\cdots\!32}a^{13}-\frac{42633044801}{37956911444452}a^{12}+\frac{52874390827683}{778116684611266}a^{11}-\frac{17910050766563}{778116684611266}a^{10}-\frac{26413020171383}{15\!\cdots\!32}a^{9}-\frac{189835156287099}{15\!\cdots\!32}a^{8}+\frac{415146860125815}{778116684611266}a^{7}-\frac{4153030449430}{55579763186519}a^{6}+\frac{224381777955779}{15\!\cdots\!32}a^{5}-\frac{663922473515261}{778116684611266}a^{4}+\frac{14\!\cdots\!55}{778116684611266}a^{3}-\frac{637857605339371}{778116684611266}a^{2}+\frac{353514029278173}{389058342305633}a-\frac{13063580099278}{29927564792741}$, $\frac{306879742045}{389058342305633}a^{17}+\frac{171559336526}{389058342305633}a^{16}+\frac{2711532894505}{15\!\cdots\!32}a^{15}-\frac{15338133706317}{15\!\cdots\!32}a^{14}+\frac{7550051916997}{15\!\cdots\!32}a^{13}-\frac{171081985885}{18978455722226}a^{12}+\frac{186953030705527}{15\!\cdots\!32}a^{11}-\frac{103228161846373}{15\!\cdots\!32}a^{10}+\frac{15089057055871}{15\!\cdots\!32}a^{9}-\frac{18521720362689}{55579763186519}a^{8}+\frac{11\!\cdots\!67}{15\!\cdots\!32}a^{7}-\frac{476768158304101}{15\!\cdots\!32}a^{6}+\frac{12\!\cdots\!61}{15\!\cdots\!32}a^{5}-\frac{11\!\cdots\!59}{778116684611266}a^{4}+\frac{133080682517390}{55579763186519}a^{3}-\frac{930399379917807}{778116684611266}a^{2}+\frac{59058396819039}{55579763186519}a-\frac{15093796513577}{29927564792741}$, $\frac{2151022395823}{47\!\cdots\!34}a^{17}+\frac{45406023912281}{66\!\cdots\!76}a^{16}+\frac{5889578454900}{23\!\cdots\!17}a^{15}-\frac{55226000698971}{33\!\cdots\!38}a^{14}+\frac{207197341669663}{66\!\cdots\!76}a^{13}-\frac{16187839324269}{16\!\cdots\!36}a^{12}+\frac{15\!\cdots\!81}{33\!\cdots\!38}a^{11}+\frac{11839411550815}{30\!\cdots\!58}a^{10}+\frac{972565915365589}{95\!\cdots\!68}a^{9}+\frac{114638832862735}{51\!\cdots\!52}a^{8}+\frac{716553017427056}{23\!\cdots\!17}a^{7}-\frac{161506547666095}{12\!\cdots\!63}a^{6}+\frac{29\!\cdots\!55}{66\!\cdots\!76}a^{5}-\frac{16\!\cdots\!22}{16\!\cdots\!19}a^{4}+\frac{19\!\cdots\!82}{16\!\cdots\!19}a^{3}+\frac{20\!\cdots\!09}{33\!\cdots\!38}a^{2}-\frac{905885241417396}{16\!\cdots\!19}a+\frac{732544120553747}{12\!\cdots\!63}$, $\frac{3658906984823}{30\!\cdots\!58}a^{17}+\frac{45581877721129}{66\!\cdots\!76}a^{16}+\frac{246281590556731}{66\!\cdots\!76}a^{15}-\frac{758375113839881}{66\!\cdots\!76}a^{14}+\frac{428078457572225}{33\!\cdots\!38}a^{13}-\frac{24340377879695}{16\!\cdots\!36}a^{12}+\frac{10\!\cdots\!05}{66\!\cdots\!76}a^{11}-\frac{76\!\cdots\!71}{66\!\cdots\!76}a^{10}+\frac{28\!\cdots\!05}{33\!\cdots\!38}a^{9}-\frac{12\!\cdots\!73}{66\!\cdots\!76}a^{8}+\frac{65\!\cdots\!27}{60\!\cdots\!16}a^{7}-\frac{53\!\cdots\!37}{95\!\cdots\!68}a^{6}+\frac{14\!\cdots\!67}{16\!\cdots\!19}a^{5}-\frac{38\!\cdots\!00}{23\!\cdots\!17}a^{4}+\frac{12\!\cdots\!69}{33\!\cdots\!38}a^{3}+\frac{14\!\cdots\!68}{23\!\cdots\!17}a^{2}-\frac{35\!\cdots\!93}{16\!\cdots\!19}a+\frac{627590184224755}{12\!\cdots\!63}$, $\frac{4886055425}{8967841714898}a^{17}+\frac{7386437739}{8967841714898}a^{16}+\frac{13686759909}{4483920857449}a^{15}-\frac{13201013377}{4483920857449}a^{14}+\frac{32930891835}{8967841714898}a^{13}-\frac{67895619029}{4483920857449}a^{12}+\frac{632648440297}{8967841714898}a^{11}+\frac{82186442}{37057197169}a^{10}+\frac{738096520065}{4483920857449}a^{9}-\frac{1072830250651}{8967841714898}a^{8}+\frac{3882069808145}{8967841714898}a^{7}-\frac{2531885562127}{8967841714898}a^{6}+\frac{4867801762657}{4483920857449}a^{5}-\frac{1459203771843}{4483920857449}a^{4}+\frac{10192366977830}{4483920857449}a^{3}-\frac{5094000891234}{4483920857449}a^{2}+\frac{6298150795456}{4483920857449}a-\frac{3137295542793}{4483920857449}$, $\frac{941253925}{51199720640068}a^{17}+\frac{136277270}{984610012309}a^{16}-\frac{37420775701}{51199720640068}a^{15}-\frac{40905729}{332465718442}a^{14}-\frac{82292642}{27061163129}a^{13}+\frac{7253629767}{624386837074}a^{12}-\frac{419638238969}{51199720640068}a^{11}+\frac{332856703617}{12799930160017}a^{10}-\frac{3459855825509}{25599860320034}a^{9}+\frac{1574426676981}{12799930160017}a^{8}-\frac{1104247390553}{51199720640068}a^{7}+\frac{764524157311}{1969220024618}a^{6}-\frac{50750616539789}{51199720640068}a^{5}+\frac{7167375802115}{12799930160017}a^{4}-\frac{5294302573529}{25599860320034}a^{3}+\frac{63498098593705}{25599860320034}a^{2}-\frac{33231624212074}{12799930160017}a+\frac{914387926211}{984610012309}$ 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:  \( 59552.7806315 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 59552.7806315 \cdot 3}{2\cdot\sqrt{2727747884710191986159616}}\cr\approx \mathstrut & 0.825486556183 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 - 12*x^15 + 18*x^14 - 24*x^13 + 155*x^12 - 198*x^11 + 198*x^10 - 382*x^9 + 1305*x^8 - 1326*x^7 + 1667*x^6 - 2772*x^5 + 5418*x^4 - 4584*x^3 + 3528*x^2 - 2340*x + 676)
 
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^16 - 12*x^15 + 18*x^14 - 24*x^13 + 155*x^12 - 198*x^11 + 198*x^10 - 382*x^9 + 1305*x^8 - 1326*x^7 + 1667*x^6 - 2772*x^5 + 5418*x^4 - 4584*x^3 + 3528*x^2 - 2340*x + 676, 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^16 - 12*x^15 + 18*x^14 - 24*x^13 + 155*x^12 - 198*x^11 + 198*x^10 - 382*x^9 + 1305*x^8 - 1326*x^7 + 1667*x^6 - 2772*x^5 + 5418*x^4 - 4584*x^3 + 3528*x^2 - 2340*x + 676);
 
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^16 - 12*x^15 + 18*x^14 - 24*x^13 + 155*x^12 - 198*x^11 + 198*x^10 - 382*x^9 + 1305*x^8 - 1326*x^7 + 1667*x^6 - 2772*x^5 + 5418*x^4 - 4584*x^3 + 3528*x^2 - 2340*x + 676);
 
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

$C_3:S_3$ (as 18T4):

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 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.1.3564.1 x3, 3.1.44.1 x3, 3.1.891.1 x3, 3.1.3564.2 x3, 6.0.139723056.1, 6.0.21296.1, 6.0.8732691.5, 6.0.139723056.2, 9.1.497972971584.1 x9

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: 9.1.497972971584.1
Minimal sibling: 9.1.497972971584.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.3.0.1}{3} }^{6}$ ${\href{/padicField/7.2.0.1}{2} }^{9}$ R ${\href{/padicField/13.2.0.1}{2} }^{9}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.2.0.1}{2} }^{9}$ ${\href{/padicField/23.3.0.1}{3} }^{6}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.2.0.1}{2} }^{9}$ ${\href{/padicField/47.3.0.1}{3} }^{6}$ ${\href{/padicField/53.3.0.1}{3} }^{6}$ ${\href{/padicField/59.3.0.1}{3} }^{6}$

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 2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
\(3\) Copy content Toggle raw display 3.9.12.1$x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$$3$$3$$12$$C_3^2$$[2]^{3}$
\(11\) Copy content Toggle raw display 11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$