Properties

Label 18.0.396...088.1
Degree $18$
Signature $[0, 9]$
Discriminant $-3.961\times 10^{28}$
Root discriminant \(38.79\)
Ramified primes $2,3,53$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_3^2$ (as 18T11)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 210*x^14 - 294*x^13 + 396*x^12 - 816*x^11 + 1678*x^10 - 2340*x^9 + 2244*x^8 - 1806*x^7 + 4167*x^6 - 9171*x^5 + 11355*x^4 - 8268*x^3 + 3658*x^2 - 936*x + 108)
 
gp: K = bnfinit(y^18 - 9*y^17 + 39*y^16 - 108*y^15 + 210*y^14 - 294*y^13 + 396*y^12 - 816*y^11 + 1678*y^10 - 2340*y^9 + 2244*y^8 - 1806*y^7 + 4167*y^6 - 9171*y^5 + 11355*y^4 - 8268*y^3 + 3658*y^2 - 936*y + 108, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 210*x^14 - 294*x^13 + 396*x^12 - 816*x^11 + 1678*x^10 - 2340*x^9 + 2244*x^8 - 1806*x^7 + 4167*x^6 - 9171*x^5 + 11355*x^4 - 8268*x^3 + 3658*x^2 - 936*x + 108);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 210*x^14 - 294*x^13 + 396*x^12 - 816*x^11 + 1678*x^10 - 2340*x^9 + 2244*x^8 - 1806*x^7 + 4167*x^6 - 9171*x^5 + 11355*x^4 - 8268*x^3 + 3658*x^2 - 936*x + 108)
 

\( x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 210 x^{14} - 294 x^{13} + 396 x^{12} - 816 x^{11} + \cdots + 108 \) 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:   \(-39606063155139137133767897088\) \(\medspace = -\,2^{12}\cdot 3^{9}\cdot 53^{12}\) 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:  \(38.79\)
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:  $2\cdot 3^{1/2}53^{2/3}\approx 48.877374517097394$
Ramified primes:   \(2\), \(3\), \(53\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{14}a^{12}+\frac{1}{14}a^{11}+\frac{3}{14}a^{10}-\frac{1}{7}a^{9}-\frac{3}{14}a^{8}-\frac{1}{2}a^{6}+\frac{3}{14}a^{5}+\frac{3}{14}a^{4}+\frac{1}{7}a^{3}+\frac{1}{14}a^{2}-\frac{1}{7}a+\frac{1}{7}$, $\frac{1}{14}a^{13}+\frac{1}{7}a^{11}+\frac{1}{7}a^{10}-\frac{1}{14}a^{9}+\frac{3}{14}a^{8}-\frac{1}{2}a^{7}-\frac{2}{7}a^{6}+\frac{3}{7}a^{4}-\frac{1}{14}a^{3}-\frac{3}{14}a^{2}+\frac{2}{7}a-\frac{1}{7}$, $\frac{1}{168}a^{14}+\frac{5}{168}a^{13}-\frac{1}{84}a^{12}-\frac{41}{168}a^{11}+\frac{13}{56}a^{10}-\frac{11}{84}a^{9}+\frac{9}{56}a^{8}-\frac{27}{56}a^{7}-\frac{19}{42}a^{6}-\frac{41}{168}a^{5}-\frac{67}{168}a^{4}-\frac{2}{21}a^{3}-\frac{13}{28}a^{2}+\frac{5}{21}a-\frac{5}{14}$, $\frac{1}{168}a^{15}-\frac{1}{56}a^{13}+\frac{5}{168}a^{12}-\frac{1}{21}a^{11}+\frac{23}{168}a^{10}+\frac{41}{168}a^{9}-\frac{1}{24}a^{7}-\frac{3}{56}a^{6}+\frac{13}{28}a^{5}-\frac{17}{168}a^{4}+\frac{25}{84}a^{3}-\frac{13}{84}a^{2}-\frac{17}{42}a-\frac{1}{14}$, $\frac{1}{99421728}a^{16}-\frac{1}{12427716}a^{15}+\frac{19135}{12427716}a^{14}-\frac{12755}{1183592}a^{13}-\frac{175169}{49710864}a^{12}+\frac{4007531}{24855432}a^{11}-\frac{1038475}{7101552}a^{10}-\frac{60397}{16570288}a^{9}-\frac{9110525}{49710864}a^{8}-\frac{775471}{24855432}a^{7}+\frac{1955825}{7101552}a^{6}+\frac{2749015}{8285144}a^{5}+\frac{26581369}{99421728}a^{4}-\frac{24684571}{49710864}a^{3}+\frac{1043509}{7101552}a^{2}-\frac{2560713}{8285144}a-\frac{3520577}{8285144}$, $\frac{1}{22767575712}a^{17}+\frac{53}{11383787856}a^{16}+\frac{268623}{135521284}a^{15}-\frac{12475363}{5691893928}a^{14}+\frac{80584481}{3794595952}a^{13}+\frac{19815309}{948648988}a^{12}-\frac{2140118065}{11383787856}a^{11}-\frac{1238407073}{11383787856}a^{10}-\frac{846444779}{11383787856}a^{9}+\frac{91016957}{1422973482}a^{8}+\frac{862089}{3794595952}a^{7}-\frac{291489886}{711486741}a^{6}+\frac{430740651}{7589191904}a^{5}+\frac{252354217}{1897297976}a^{4}+\frac{638400499}{1626255408}a^{3}+\frac{317688842}{711486741}a^{2}+\frac{672736397}{1897297976}a+\frac{72145067}{948648988}$ 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

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:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{19092757}{203281926} a^{17} + \frac{324576869}{406563852} a^{16} - \frac{331895264}{101640963} a^{15} + \frac{866330135}{101640963} a^{14} - \frac{1576515436}{101640963} a^{13} + \frac{676919711}{33880321} a^{12} - \frac{2786618690}{101640963} a^{11} + \frac{4282069165}{67760642} a^{10} - \frac{12845807312}{101640963} a^{9} + \frac{16008645040}{101640963} a^{8} - \frac{13593525280}{101640963} a^{7} + \frac{10647508315}{101640963} a^{6} - \frac{69237356587}{203281926} a^{5} + \frac{93809561563}{135521284} a^{4} - \frac{73745502826}{101640963} a^{3} + \frac{14329182505}{33880321} a^{2} - \frac{14326293356}{101640963} a + \frac{755635754}{33880321} \)  (order $6$) 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:   $\frac{18052201}{271042568}a^{17}-\frac{13351806401}{22767575712}a^{16}+\frac{14000216869}{5691893928}a^{15}-\frac{37187594375}{5691893928}a^{14}+\frac{9797462077}{813127704}a^{13}-\frac{178422686125}{11383787856}a^{12}+\frac{29607140215}{1422973482}a^{11}-\frac{541212869683}{11383787856}a^{10}+\frac{1117572658109}{11383787856}a^{9}-\frac{1420460949137}{11383787856}a^{8}+\frac{591797092037}{5691893928}a^{7}-\frac{129267557411}{1626255408}a^{6}+\frac{1408841417483}{5691893928}a^{5}-\frac{12532207040305}{22767575712}a^{4}+\frac{6553259978155}{11383787856}a^{3}-\frac{3716453308115}{11383787856}a^{2}+\frac{570089597723}{5691893928}a-\frac{24574851943}{1897297976}$, $\frac{18052201}{271042568}a^{17}-\frac{12426736627}{22767575712}a^{16}+\frac{12150077321}{5691893928}a^{15}-\frac{15023448455}{2845946964}a^{14}+\frac{3641056741}{406563852}a^{13}-\frac{118321592615}{11383787856}a^{12}+\frac{82840653041}{5691893928}a^{11}-\frac{437788700171}{11383787856}a^{10}+\frac{854558731639}{11383787856}a^{9}-\frac{922864355785}{11383787856}a^{8}+\frac{158356731929}{2845946964}a^{7}-\frac{69239075221}{1626255408}a^{6}+\frac{311505089407}{1422973482}a^{5}-\frac{9487595289983}{22767575712}a^{4}+\frac{3880924426457}{11383787856}a^{3}-\frac{190718617939}{1626255408}a^{2}+\frac{30817502761}{5691893928}a+\frac{6826228495}{1897297976}$, $\frac{574816099}{22767575712}a^{17}-\frac{1157228213}{5691893928}a^{16}+\frac{4411107301}{5691893928}a^{15}-\frac{10556978095}{5691893928}a^{14}+\frac{11435099181}{3794595952}a^{13}-\frac{1329198257}{406563852}a^{12}+\frac{53712211549}{11383787856}a^{11}-\frac{154989937915}{11383787856}a^{10}+\frac{294108759683}{11383787856}a^{9}-\frac{142021045021}{5691893928}a^{8}+\frac{167148777151}{11383787856}a^{7}-\frac{35235971287}{2845946964}a^{6}+\frac{605797898249}{7589191904}a^{5}-\frac{1647149836345}{11383787856}a^{4}+\frac{1067491597409}{11383787856}a^{3}-\frac{109726780357}{5691893928}a^{2}-\frac{29455892609}{5691893928}a+\frac{594043477}{237162247}$, $\frac{1010766873}{7589191904}a^{17}-\frac{8799876017}{7589191904}a^{16}+\frac{2294514919}{474324494}a^{15}-\frac{3479270907}{271042568}a^{14}+\frac{90020568109}{3794595952}a^{13}-\frac{117836386025}{3794595952}a^{12}+\frac{158453494573}{3794595952}a^{11}-\frac{178956504365}{1897297976}a^{10}+\frac{365444349665}{1897297976}a^{9}-\frac{935290763761}{3794595952}a^{8}+\frac{114249536051}{542085136}a^{7}-\frac{618018398541}{3794595952}a^{6}+\frac{3764339775149}{7589191904}a^{5}-\frac{8090322130983}{7589191904}a^{4}+\frac{311154975305}{271042568}a^{3}-\frac{2579288411145}{3794595952}a^{2}+\frac{53307270197}{237162247}a-\frac{64438780949}{1897297976}$, $\frac{2259951167}{11383787856}a^{17}-\frac{12413813567}{7589191904}a^{16}+\frac{9225034619}{1422973482}a^{15}-\frac{11664670355}{711486741}a^{14}+\frac{81981057125}{2845946964}a^{13}-\frac{134714700809}{3794595952}a^{12}+\frac{142579447427}{2845946964}a^{11}-\frac{1391752931749}{11383787856}a^{10}+\frac{900902760963}{3794595952}a^{9}-\frac{1048880765529}{3794595952}a^{8}+\frac{632735905589}{2845946964}a^{7}-\frac{2017749601459}{11383787856}a^{6}+\frac{7777895262589}{11383787856}a^{5}-\frac{1401304653293}{1084170272}a^{4}+\frac{13917975368569}{11383787856}a^{3}-\frac{7347574626659}{11383787856}a^{2}+\frac{1066543962653}{5691893928}a-\frac{45381468835}{1897297976}$, $\frac{587}{1775388}a^{16}-\frac{1174}{443847}a^{15}+\frac{98997}{8285144}a^{14}-\frac{132631}{3550776}a^{13}+\frac{49551}{591796}a^{12}-\frac{162323}{1183592}a^{11}+\frac{648979}{3550776}a^{10}-\frac{411419}{1775388}a^{9}+\frac{7618889}{24855432}a^{8}-\frac{9582595}{24855432}a^{7}+\frac{7199}{147949}a^{6}+\frac{2620519}{3550776}a^{5}-\frac{1943159}{1183592}a^{4}+\frac{268904}{147949}a^{3}-\frac{16054967}{12427716}a^{2}+\frac{1673081}{3106929}a-\frac{230989}{2071286}$, $\frac{8587575379}{2845946964}a^{17}-\frac{37953455195}{1422973482}a^{16}+\frac{319363846153}{2845946964}a^{15}-\frac{849630485663}{2845946964}a^{14}+\frac{261086611239}{474324494}a^{13}-\frac{2032041003943}{2845946964}a^{12}+\frac{892367949841}{948648988}a^{11}-\frac{510930414359}{237162247}a^{10}+\frac{1823009253959}{406563852}a^{9}-\frac{2319326757509}{406563852}a^{8}+\frac{3326653918169}{711486741}a^{7}-\frac{9977712574939}{2845946964}a^{6}+\frac{5306243396413}{474324494}a^{5}-\frac{17915662003094}{711486741}a^{4}+\frac{895931058910}{33880321}a^{3}-\frac{6784262751061}{474324494}a^{2}+\frac{935000987127}{237162247}a-\frac{105381992872}{237162247}$, $\frac{8726707223}{474324494}a^{17}-\frac{31437248485}{203281926}a^{16}+\frac{889198814341}{1422973482}a^{15}-\frac{218155364465}{135521284}a^{14}+\frac{8208504216809}{2845946964}a^{13}-\frac{739847386277}{203281926}a^{12}+\frac{14313745823011}{2845946964}a^{11}-\frac{1614907477431}{135521284}a^{10}+\frac{33632468144441}{1422973482}a^{9}-\frac{81380008876021}{2845946964}a^{8}+\frac{9558182134741}{406563852}a^{7}-\frac{4369929042505}{237162247}a^{6}+\frac{185355090563329}{2845946964}a^{5}-\frac{52670091974753}{406563852}a^{4}+\frac{184346207074783}{1422973482}a^{3}-\frac{16803271138890}{237162247}a^{2}+\frac{15463603109582}{711486741}a-\frac{716191620374}{237162247}$ 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:  \( 373923084.8527347 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 373923084.8527347 \cdot 1}{6\cdot\sqrt{39606063155139137133767897088}}\cr\approx \mathstrut & 4.77935186174263 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 210*x^14 - 294*x^13 + 396*x^12 - 816*x^11 + 1678*x^10 - 2340*x^9 + 2244*x^8 - 1806*x^7 + 4167*x^6 - 9171*x^5 + 11355*x^4 - 8268*x^3 + 3658*x^2 - 936*x + 108)
 
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 - 9*x^17 + 39*x^16 - 108*x^15 + 210*x^14 - 294*x^13 + 396*x^12 - 816*x^11 + 1678*x^10 - 2340*x^9 + 2244*x^8 - 1806*x^7 + 4167*x^6 - 9171*x^5 + 11355*x^4 - 8268*x^3 + 3658*x^2 - 936*x + 108, 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 - 9*x^17 + 39*x^16 - 108*x^15 + 210*x^14 - 294*x^13 + 396*x^12 - 816*x^11 + 1678*x^10 - 2340*x^9 + 2244*x^8 - 1806*x^7 + 4167*x^6 - 9171*x^5 + 11355*x^4 - 8268*x^3 + 3658*x^2 - 936*x + 108);
 
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 - 9*x^17 + 39*x^16 - 108*x^15 + 210*x^14 - 294*x^13 + 396*x^12 - 816*x^11 + 1678*x^10 - 2340*x^9 + 2244*x^8 - 1806*x^7 + 4167*x^6 - 9171*x^5 + 11355*x^4 - 8268*x^3 + 3658*x^2 - 936*x + 108);
 
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

$S_3^2$ (as 18T11):

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 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.33708.1, 3.1.8427.1 x3, 6.0.3408687792.2, 6.0.213042987.1, 9.3.114900048092736.1 x3

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: data not computed
Degree 6 sibling: 6.0.1617984.1
Degree 9 sibling: 9.3.114900048092736.1
Degree 12 sibling: deg 12
Degree 18 siblings: 18.6.2534788041928904776561145413632.1, deg 18
Minimal sibling: 6.0.1617984.1

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 ${\href{/padicField/5.2.0.1}{2} }^{9}$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ R ${\href{/padicField/59.6.0.1}{6} }^{3}$

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 2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
\(3\) Copy content Toggle raw display 3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(53\) Copy content Toggle raw display 53.6.4.1$x^{6} + 147 x^{5} + 7209 x^{4} + 118343 x^{3} + 22209 x^{2} + 381711 x + 6222632$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
53.6.4.1$x^{6} + 147 x^{5} + 7209 x^{4} + 118343 x^{3} + 22209 x^{2} + 381711 x + 6222632$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
53.6.4.1$x^{6} + 147 x^{5} + 7209 x^{4} + 118343 x^{3} + 22209 x^{2} + 381711 x + 6222632$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$