Properties

Label 18.18.454...613.1
Degree $18$
Signature $[18, 0]$
Discriminant $4.546\times 10^{31}$
Root discriminant \(57.38\)
Ramified primes $3,13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3^2 : C_4$ (as 18T10)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 36*x^16 + 486*x^14 - 3345*x^12 + 13113*x^10 - 30537*x^8 + 42276*x^6 - 33462*x^4 + 13689*x^2 - 2197)
 
gp: K = bnfinit(y^18 - 36*y^16 + 486*y^14 - 3345*y^12 + 13113*y^10 - 30537*y^8 + 42276*y^6 - 33462*y^4 + 13689*y^2 - 2197, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 36*x^16 + 486*x^14 - 3345*x^12 + 13113*x^10 - 30537*x^8 + 42276*x^6 - 33462*x^4 + 13689*x^2 - 2197);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 36*x^16 + 486*x^14 - 3345*x^12 + 13113*x^10 - 30537*x^8 + 42276*x^6 - 33462*x^4 + 13689*x^2 - 2197)
 

\( x^{18} - 36 x^{16} + 486 x^{14} - 3345 x^{12} + 13113 x^{10} - 30537 x^{8} + 42276 x^{6} - 33462 x^{4} + 13689 x^{2} - 2197 \) 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:   \(45459928664503948293335446409613\) \(\medspace = 3^{36}\cdot 13^{13}\) 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:  \(57.38\)
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:  $3^{37/18}13^{3/4}\approx 65.49479091539757$
Ramified primes:   \(3\), \(13\) 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:  \(\Q(\sqrt{13}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{26}a^{12}+\frac{3}{26}a^{10}+\frac{5}{26}a^{8}-\frac{2}{13}a^{6}-\frac{2}{13}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{26}a^{13}+\frac{3}{26}a^{11}+\frac{5}{26}a^{9}-\frac{2}{13}a^{7}-\frac{2}{13}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{26}a^{14}-\frac{2}{13}a^{10}+\frac{7}{26}a^{8}+\frac{4}{13}a^{6}-\frac{1}{2}a^{5}-\frac{1}{26}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{338}a^{15}+\frac{3}{338}a^{13}+\frac{57}{338}a^{11}-\frac{28}{169}a^{9}-\frac{41}{169}a^{7}-\frac{1}{2}a^{6}-\frac{9}{26}a^{5}+\frac{1}{26}a^{3}$, $\frac{1}{1626794}a^{16}+\frac{6781}{813397}a^{14}+\frac{25719}{1626794}a^{12}-\frac{49831}{813397}a^{10}-\frac{446775}{1626794}a^{8}-\frac{1}{2}a^{7}-\frac{15577}{125138}a^{6}-\frac{1}{2}a^{5}-\frac{15584}{62569}a^{4}-\frac{1}{2}a^{3}+\frac{668}{4813}a^{2}+\frac{2747}{9626}$, $\frac{1}{1626794}a^{17}-\frac{877}{1626794}a^{15}-\frac{8799}{813397}a^{13}-\frac{54644}{813397}a^{11}+\frac{361809}{1626794}a^{9}+\frac{84050}{813397}a^{7}-\frac{1}{2}a^{6}-\frac{26355}{125138}a^{5}-\frac{1}{2}a^{4}-\frac{29820}{62569}a^{3}-\frac{1}{2}a^{2}+\frac{2747}{9626}a$ 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:  $17$
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:   $\frac{11689}{813397}a^{17}-\frac{1217}{1626794}a^{16}-\frac{31436}{62569}a^{15}+\frac{75831}{1626794}a^{14}+\frac{5274833}{813397}a^{13}-\frac{1642317}{1626794}a^{12}-\frac{68437991}{1626794}a^{11}+\frac{16235303}{1626794}a^{10}+\frac{124581993}{813397}a^{9}-\frac{82590515}{1626794}a^{8}-\frac{529292001}{1626794}a^{7}+\frac{17282285}{125138}a^{6}+\frac{49659913}{125138}a^{5}-\frac{24777121}{125138}a^{4}-\frac{31535165}{125138}a^{3}+\frac{664635}{4813}a^{2}+\frac{596319}{9626}a-\frac{172300}{4813}$, $\frac{28142}{813397}a^{17}+\frac{41775}{813397}a^{16}-\frac{148381}{125138}a^{15}-\frac{2835095}{1626794}a^{14}+\frac{24022373}{1626794}a^{13}+\frac{17370539}{813397}a^{12}-\frac{73416390}{813397}a^{11}-\frac{207493815}{1626794}a^{10}+\frac{485429221}{1626794}a^{9}+\frac{331985344}{813397}a^{8}-\frac{443733157}{813397}a^{7}-\frac{44560703}{62569}a^{6}+\frac{33518622}{62569}a^{5}+\frac{40807865}{62569}a^{4}-\frac{32356839}{125138}a^{3}-\frac{1357414}{4813}a^{2}+\frac{230692}{4813}a+\frac{211338}{4813}$, $\frac{5488}{813397}a^{17}-\frac{6001}{813397}a^{16}-\frac{418683}{1626794}a^{15}+\frac{204414}{813397}a^{14}+\frac{3046463}{813397}a^{13}-\frac{5032081}{1626794}a^{12}-\frac{45161477}{1626794}a^{11}+\frac{15053720}{813397}a^{10}+\frac{14351229}{125138}a^{9}-\frac{47912729}{813397}a^{8}-\frac{217118218}{813397}a^{7}+\frac{12681237}{125138}a^{6}+\frac{20878017}{62569}a^{5}-\frac{11649923}{125138}a^{4}-\frac{12452299}{62569}a^{3}+\frac{440227}{9626}a^{2}+\frac{217805}{4813}a-\frac{82265}{9626}$, $\frac{158515}{1626794}a^{17}+\frac{138207}{1626794}a^{16}-\frac{5620547}{1626794}a^{15}-\frac{4896581}{1626794}a^{14}+\frac{73923315}{1626794}a^{13}+\frac{64321875}{1626794}a^{12}-\frac{243985538}{813397}a^{11}-\frac{211944390}{813397}a^{10}+\frac{894470998}{813397}a^{9}+\frac{1550871839}{1626794}a^{8}-\frac{1866815518}{813397}a^{7}-\frac{248461851}{125138}a^{6}+\frac{330212227}{125138}a^{5}+\frac{285309051}{125138}a^{4}-\frac{93370005}{62569}a^{3}-\frac{6214273}{4813}a^{2}+\frac{1455417}{4813}a+\frac{1266007}{4813}$, $\frac{36917}{1626794}a^{17}-\frac{1255351}{1626794}a^{15}+\frac{7726365}{813397}a^{13}-\frac{46590446}{813397}a^{11}+\frac{900444}{4813}a^{9}-\frac{277214229}{813397}a^{7}+\frac{21237570}{62569}a^{5}-\frac{10556798}{62569}a^{3}+\frac{1}{2}a^{2}+\frac{152154}{4813}a-\frac{1}{2}$, $\frac{4611}{1626794}a^{16}-\frac{79878}{813397}a^{14}+\frac{1012131}{813397}a^{12}-\frac{6336842}{813397}a^{10}+\frac{21651274}{813397}a^{8}-\frac{3201219}{62569}a^{6}+\frac{3484878}{62569}a^{4}-\frac{168627}{4813}a^{2}+\frac{1}{2}a+\frac{49837}{4813}$, $\frac{6593}{813397}a^{17}-\frac{83071}{1626794}a^{16}-\frac{227859}{813397}a^{15}+\frac{1443343}{813397}a^{14}+\frac{2876738}{813397}a^{13}-\frac{36812547}{1626794}a^{12}-\frac{2757973}{125138}a^{11}+\frac{233461999}{1626794}a^{10}+\frac{60775867}{813397}a^{9}-\frac{408047586}{813397}a^{8}-\frac{228827253}{1626794}a^{7}+\frac{124543853}{125138}a^{6}+\frac{8899648}{62569}a^{5}-\frac{68429445}{62569}a^{4}-\frac{4417193}{62569}a^{3}+\frac{2895075}{4813}a^{2}+\frac{138881}{9626}a-\frac{1176453}{9626}$, $\frac{8575}{1626794}a^{17}+\frac{383}{62569}a^{16}-\frac{147981}{813397}a^{15}-\frac{13420}{62569}a^{14}+\frac{1864607}{813397}a^{13}+\frac{347681}{125138}a^{12}-\frac{23179965}{1626794}a^{11}-\frac{2264209}{125138}a^{10}+\frac{39124593}{813397}a^{9}+\frac{316683}{4813}a^{8}-\frac{145595559}{1626794}a^{7}-\frac{17179999}{125138}a^{6}+\frac{5414770}{62569}a^{5}+\frac{9924493}{62569}a^{4}-\frac{2225042}{62569}a^{3}-\frac{860771}{9626}a^{2}+\frac{12384}{4813}a+\frac{165989}{9626}$, $\frac{5488}{813397}a^{17}+\frac{6001}{813397}a^{16}-\frac{418683}{1626794}a^{15}-\frac{204414}{813397}a^{14}+\frac{3046463}{813397}a^{13}+\frac{5032081}{1626794}a^{12}-\frac{45161477}{1626794}a^{11}-\frac{15053720}{813397}a^{10}+\frac{14351229}{125138}a^{9}+\frac{47912729}{813397}a^{8}-\frac{217118218}{813397}a^{7}-\frac{12681237}{125138}a^{6}+\frac{20878017}{62569}a^{5}+\frac{11649923}{125138}a^{4}-\frac{12452299}{62569}a^{3}-\frac{440227}{9626}a^{2}+\frac{217805}{4813}a+\frac{82265}{9626}$, $\frac{4413}{125138}a^{17}+\frac{97393}{1626794}a^{16}-\frac{154565}{125138}a^{15}-\frac{3303881}{1626794}a^{14}+\frac{997588}{62569}a^{13}+\frac{40507933}{1626794}a^{12}-\frac{12851989}{125138}a^{11}-\frac{121409958}{813397}a^{10}+\frac{1757983}{4813}a^{9}+\frac{785719011}{1626794}a^{8}-\frac{45960532}{62569}a^{7}-\frac{54131546}{62569}a^{6}+\frac{50450185}{62569}a^{5}+\frac{104210713}{125138}a^{4}-\frac{4160473}{9626}a^{3}-\frac{1871054}{4813}a^{2}+\frac{414210}{4813}a+\frac{648095}{9626}$, $\frac{11127}{1626794}a^{17}+\frac{13151}{813397}a^{16}-\frac{193734}{813397}a^{15}-\frac{468340}{813397}a^{14}+\frac{2463829}{813397}a^{13}+\frac{12416341}{1626794}a^{12}-\frac{15312464}{813397}a^{11}-\frac{41505366}{813397}a^{10}+\frac{50195004}{813397}a^{9}+\frac{310162373}{1626794}a^{8}-\frac{12830981}{125138}a^{7}-\frac{51123315}{125138}a^{6}+\frac{8089845}{125138}a^{5}+\frac{30446189}{62569}a^{4}+\frac{181209}{9626}a^{3}-\frac{2801407}{9626}a^{2}-\frac{246957}{9626}a+\frac{658219}{9626}$, $\frac{13388}{813397}a^{17}+\frac{198}{62569}a^{16}-\frac{493295}{813397}a^{15}-\frac{5191}{62569}a^{14}+\frac{13681637}{1626794}a^{13}+\frac{33899}{62569}a^{12}-\frac{95755749}{1626794}a^{11}+\frac{72643}{125138}a^{10}+\frac{185569460}{813397}a^{9}-\frac{1066996}{62569}a^{8}-\frac{808381659}{1626794}a^{7}+\frac{3727167}{62569}a^{6}+\frac{36511708}{62569}a^{5}-\frac{5056998}{62569}a^{4}-\frac{41171333}{125138}a^{3}+\frac{404243}{9626}a^{2}+\frac{641535}{9626}a-\frac{71233}{9626}$, $\frac{1471}{1626794}a^{17}+\frac{24316}{813397}a^{16}-\frac{45815}{813397}a^{15}-\frac{1619439}{1626794}a^{14}+\frac{152725}{125138}a^{13}+\frac{19220781}{1626794}a^{12}-\frac{19621423}{1626794}a^{11}-\frac{54706559}{813397}a^{10}+\frac{49633812}{813397}a^{9}+\frac{163317091}{813397}a^{8}-\frac{264515255}{1626794}a^{7}-\frac{19951556}{62569}a^{6}+\frac{27506805}{125138}a^{5}+\frac{2520587}{9626}a^{4}-\frac{16704999}{125138}a^{3}-\frac{1018691}{9626}a^{2}+\frac{267445}{9626}a+\frac{161305}{9626}$, $\frac{4865}{1626794}a^{17}+\frac{22305}{1626794}a^{16}-\frac{175555}{1626794}a^{15}-\frac{771533}{1626794}a^{14}+\frac{1183685}{813397}a^{13}+\frac{9790467}{1626794}a^{12}-\frac{8029902}{813397}a^{11}-\frac{62015357}{1626794}a^{10}+\frac{59979657}{1626794}a^{9}+\frac{109175748}{813397}a^{8}-\frac{61538622}{813397}a^{7}-\frac{33964669}{125138}a^{6}+\frac{4849720}{62569}a^{5}+\frac{19202423}{62569}a^{4}-\frac{1873111}{62569}a^{3}-\frac{848396}{4813}a^{2}+\frac{4040}{4813}a+\frac{191286}{4813}$, $\frac{39401}{813397}a^{17}-\frac{21981}{1626794}a^{16}-\frac{1344977}{813397}a^{15}+\frac{785791}{1626794}a^{14}+\frac{33284363}{1626794}a^{13}-\frac{5202439}{813397}a^{12}-\frac{100803624}{813397}a^{11}+\frac{68392511}{1626794}a^{10}+\frac{658485135}{1626794}a^{9}-\frac{122050179}{813397}a^{8}-\frac{592842052}{813397}a^{7}+\frac{18398301}{62569}a^{6}+\frac{43991841}{62569}a^{5}-\frac{19015095}{62569}a^{4}-\frac{41455539}{125138}a^{3}+\frac{718292}{4813}a^{2}+\frac{553101}{9626}a-\frac{248185}{9626}$, $\frac{10737}{813397}a^{17}+\frac{53375}{1626794}a^{16}-\frac{27190}{62569}a^{15}-\frac{1866081}{1626794}a^{14}+\frac{4109330}{813397}a^{13}+\frac{24014261}{1626794}a^{12}-\frac{22579200}{813397}a^{11}-\frac{77006443}{813397}a^{10}+\frac{63129786}{813397}a^{9}+\frac{272610383}{813397}a^{8}-\frac{173878729}{1626794}a^{7}-\frac{6484645}{9626}a^{6}+\frac{3635702}{62569}a^{5}+\frac{93773877}{125138}a^{4}+\frac{154547}{62569}a^{3}-\frac{1997599}{4813}a^{2}-\frac{33216}{4813}a+\frac{400832}{4813}$, $\frac{33589}{1626794}a^{17}+\frac{17963}{813397}a^{16}-\frac{1108983}{1626794}a^{15}-\frac{592601}{813397}a^{14}+\frac{12956343}{1626794}a^{13}+\frac{6926060}{813397}a^{12}-\frac{71756549}{1626794}a^{11}-\frac{38546782}{813397}a^{10}+\frac{203731043}{1626794}a^{9}+\frac{111053435}{813397}a^{8}-\frac{145815795}{813397}a^{7}-\frac{12702590}{62569}a^{6}+\frac{6970502}{62569}a^{5}+\frac{17960185}{125138}a^{4}-\frac{1285423}{125138}a^{3}-\frac{349449}{9626}a^{2}-\frac{92487}{9626}a-\frac{11469}{9626}$ 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:  \( 40148447797.4 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 40148447797.4 \cdot 1}{2\cdot\sqrt{45459928664503948293335446409613}}\cr\approx \mathstrut & 0.780484550563 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 36*x^16 + 486*x^14 - 3345*x^12 + 13113*x^10 - 30537*x^8 + 42276*x^6 - 33462*x^4 + 13689*x^2 - 2197)
 
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 - 36*x^16 + 486*x^14 - 3345*x^12 + 13113*x^10 - 30537*x^8 + 42276*x^6 - 33462*x^4 + 13689*x^2 - 2197, 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 - 36*x^16 + 486*x^14 - 3345*x^12 + 13113*x^10 - 30537*x^8 + 42276*x^6 - 33462*x^4 + 13689*x^2 - 2197);
 
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 - 36*x^16 + 486*x^14 - 3345*x^12 + 13113*x^10 - 30537*x^8 + 42276*x^6 - 33462*x^4 + 13689*x^2 - 2197);
 
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

$C_3^2:C_4$ (as 18T10):

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 6 conjugacy class representatives for $C_3^2 : C_4$
Character table for $C_3^2 : C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), 6.6.187388721.1 x2, 6.6.1686498489.1 x2, 9.9.1870004703089601.1

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: data not computed
Degree 6 siblings: 6.6.1686498489.1, 6.6.187388721.1
Degree 9 sibling: 9.9.1870004703089601.1
Degree 12 siblings: deg 12, deg 12
Minimal sibling: 6.6.187388721.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}{,}\,{\href{/padicField/2.2.0.1}{2} }$ R ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }$ R ${\href{/padicField/17.3.0.1}{3} }^{6}$ ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.3.0.1}{3} }^{6}$ ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
\(3\) Copy content Toggle raw display 3.9.18.2$x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 3$$9$$1$$18$$C_3^2:C_2$$[3/2, 5/2]_{2}$
3.9.18.2$x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 3$$9$$1$$18$$C_3^2:C_2$$[3/2, 5/2]_{2}$
\(13\) Copy content Toggle raw display 13.2.1.1$x^{2} + 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} + 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} + 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} + 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} + 13$$4$$1$$3$$C_4$$[\ ]_{4}$