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Properties

Label 18.18.129...849.1
Degree $18$
Signature $[18, 0]$
Discriminant $1.292\times 10^{31}$
Root discriminant \(53.51\)
Ramified primes $11,19$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 56*x^16 + 56*x^15 + 1312*x^14 - 1312*x^13 - 16643*x^12 + 16643*x^11 + 123406*x^10 - 123406*x^9 - 536825*x^8 + 536825*x^7 + 1291507*x^6 - 1291507*x^5 - 1450991*x^4 + 1450991*x^3 + 418894*x^2 - 418894*x + 44917)
 
gp: K = bnfinit(y^18 - y^17 - 56*y^16 + 56*y^15 + 1312*y^14 - 1312*y^13 - 16643*y^12 + 16643*y^11 + 123406*y^10 - 123406*y^9 - 536825*y^8 + 536825*y^7 + 1291507*y^6 - 1291507*y^5 - 1450991*y^4 + 1450991*y^3 + 418894*y^2 - 418894*y + 44917, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 - 56*x^16 + 56*x^15 + 1312*x^14 - 1312*x^13 - 16643*x^12 + 16643*x^11 + 123406*x^10 - 123406*x^9 - 536825*x^8 + 536825*x^7 + 1291507*x^6 - 1291507*x^5 - 1450991*x^4 + 1450991*x^3 + 418894*x^2 - 418894*x + 44917);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - x^17 - 56*x^16 + 56*x^15 + 1312*x^14 - 1312*x^13 - 16643*x^12 + 16643*x^11 + 123406*x^10 - 123406*x^9 - 536825*x^8 + 536825*x^7 + 1291507*x^6 - 1291507*x^5 - 1450991*x^4 + 1450991*x^3 + 418894*x^2 - 418894*x + 44917)
 

\( x^{18} - x^{17} - 56 x^{16} + 56 x^{15} + 1312 x^{14} - 1312 x^{13} - 16643 x^{12} + 16643 x^{11} + \cdots + 44917 \) 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:   \(12922465537100419689226617716849\) \(\medspace = 11^{9}\cdot 19^{17}\) 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:  \(53.51\)
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:  $11^{1/2}19^{17/18}\approx 53.50668890125934$
Ramified primes:   \(11\), \(19\) 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:  \(\Q(\sqrt{209}) \)
$\card{ \Gal(K/\Q) }$:  $18$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(209=11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{209}(1,·)$, $\chi_{209}(65,·)$, $\chi_{209}(10,·)$, $\chi_{209}(45,·)$, $\chi_{209}(208,·)$, $\chi_{209}(21,·)$, $\chi_{209}(23,·)$, $\chi_{209}(100,·)$, $\chi_{209}(32,·)$, $\chi_{209}(144,·)$, $\chi_{209}(98,·)$, $\chi_{209}(164,·)$, $\chi_{209}(199,·)$, $\chi_{209}(109,·)$, $\chi_{209}(111,·)$, $\chi_{209}(177,·)$, $\chi_{209}(186,·)$, $\chi_{209}(188,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{15467}a^{10}-\frac{6479}{15467}a^{9}-\frac{30}{15467}a^{8}+\frac{4796}{15467}a^{7}+\frac{315}{15467}a^{6}+\frac{3237}{15467}a^{5}-\frac{1350}{15467}a^{4}+\frac{4677}{15467}a^{3}+\frac{2025}{15467}a^{2}-\frac{5756}{15467}a-\frac{486}{15467}$, $\frac{1}{15467}a^{11}-\frac{33}{15467}a^{9}-\frac{3970}{15467}a^{8}+\frac{396}{15467}a^{7}+\frac{2478}{15467}a^{6}-\frac{2079}{15467}a^{5}-\frac{3118}{15467}a^{4}+\frac{4455}{15467}a^{3}-\frac{1797}{15467}a^{2}-\frac{2673}{15467}a+\frac{6474}{15467}$, $\frac{1}{15467}a^{12}-\frac{1239}{15467}a^{9}-\frac{594}{15467}a^{8}+\frac{6076}{15467}a^{7}-\frac{7151}{15467}a^{6}-\frac{4566}{15467}a^{5}+\frac{6306}{15467}a^{4}-\frac{2126}{15467}a^{3}+\frac{2284}{15467}a^{2}+\frac{2130}{15467}a-\frac{571}{15467}$, $\frac{1}{15467}a^{13}-\frac{702}{15467}a^{9}-\frac{160}{15467}a^{8}-\frac{4235}{15467}a^{7}-\frac{956}{15467}a^{6}-\frac{4471}{15467}a^{5}-\frac{4340}{15467}a^{4}-\frac{3038}{15467}a^{3}+\frac{5451}{15467}a^{2}-\frac{1968}{15467}a+\frac{1059}{15467}$, $\frac{1}{15467}a^{14}-\frac{1120}{15467}a^{9}+\frac{5639}{15467}a^{8}-\frac{5970}{15467}a^{7}+\frac{121}{15467}a^{6}-\frac{5615}{15467}a^{5}-\frac{7251}{15467}a^{4}-\frac{5766}{15467}a^{3}-\frac{3382}{15467}a^{2}-\frac{2766}{15467}a-\frac{898}{15467}$, $\frac{1}{15467}a^{15}+\frac{3182}{15467}a^{9}+\frac{6831}{15467}a^{8}+\frac{4592}{15467}a^{7}+\frac{6911}{15467}a^{6}-\frac{1089}{15467}a^{5}-\frac{2000}{15467}a^{4}+\frac{7012}{15467}a^{3}+\frac{7052}{15467}a^{2}+\frac{2121}{15467}a-\frac{2975}{15467}$, $\frac{1}{15467}a^{16}+\frac{5498}{15467}a^{9}+\frac{7250}{15467}a^{8}-\frac{3499}{15467}a^{7}+\frac{1936}{15467}a^{6}-\frac{1112}{15467}a^{5}+\frac{2886}{15467}a^{4}+\frac{4092}{15467}a^{3}-\frac{7157}{15467}a^{2}-\frac{311}{15467}a-\frac{248}{15467}$, $\frac{1}{15467}a^{17}-\frac{7176}{15467}a^{9}+\frac{6771}{15467}a^{8}+\frac{4763}{15467}a^{7}-\frac{678}{15467}a^{6}-\frac{7090}{15467}a^{5}+\frac{2232}{15467}a^{4}+\frac{318}{15467}a^{3}+\frac{2479}{15467}a^{2}+\frac{758}{15467}a-\frac{3763}{15467}$ 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:   $\frac{2}{15467}a^{16}-\frac{96}{15467}a^{14}+\frac{16}{15467}a^{13}+\frac{1859}{15467}a^{12}-\frac{624}{15467}a^{11}-\frac{18466}{15467}a^{10}+\frac{9391}{15467}a^{9}+\frac{98382}{15467}a^{8}-\frac{68069}{15467}a^{7}-\frac{263371}{15467}a^{6}+\frac{240045}{15467}a^{5}+\frac{268641}{15467}a^{4}-\frac{354803}{15467}a^{3}+\frac{32157}{15467}a^{2}+\frac{108396}{15467}a-\frac{45553}{15467}$, $\frac{16}{15467}a^{13}-\frac{624}{15467}a^{11}+\frac{9360}{15467}a^{9}-\frac{67392}{15467}a^{7}+\frac{599}{15467}a^{6}+\frac{235872}{15467}a^{5}-\frac{10782}{15467}a^{4}-\frac{353808}{15467}a^{3}+\frac{48519}{15467}a^{2}+\frac{151632}{15467}a-\frac{32346}{15467}$, $\frac{35}{15467}a^{11}-\frac{1155}{15467}a^{9}+\frac{253}{15467}a^{8}+\frac{13860}{15467}a^{7}-\frac{6072}{15467}a^{6}-\frac{72765}{15467}a^{5}+\frac{45540}{15467}a^{4}+\frac{155925}{15467}a^{3}-\frac{109296}{15467}a^{2}-\frac{93555}{15467}a+\frac{25519}{15467}$, $\frac{35}{15467}a^{11}-\frac{1155}{15467}a^{9}+\frac{253}{15467}a^{8}+\frac{13860}{15467}a^{7}-\frac{6072}{15467}a^{6}-\frac{72765}{15467}a^{5}+\frac{45540}{15467}a^{4}+\frac{155925}{15467}a^{3}-\frac{109296}{15467}a^{2}-\frac{93555}{15467}a+\frac{40986}{15467}$, $\frac{13}{15467}a^{12}-\frac{468}{15467}a^{10}+\frac{6318}{15467}a^{8}-\frac{160}{15467}a^{7}-\frac{39312}{15467}a^{6}+\frac{3360}{15467}a^{5}+\frac{110565}{15467}a^{4}-\frac{20160}{15467}a^{3}-\frac{113724}{15467}a^{2}+\frac{30240}{15467}a+\frac{18954}{15467}$, $\frac{1}{15467}a^{17}-\frac{51}{15467}a^{15}-\frac{1}{15467}a^{14}+\frac{1071}{15467}a^{13}+\frac{42}{15467}a^{12}-\frac{11969}{15467}a^{11}-\frac{693}{15467}a^{10}+\frac{76890}{15467}a^{9}+\frac{5417}{15467}a^{8}-\frac{286506}{15467}a^{7}-\frac{17742}{15467}a^{6}+\frac{592192}{15467}a^{5}+\frac{2088}{15467}a^{4}-\frac{585888}{15467}a^{3}+\frac{71774}{15467}a^{2}+\frac{156537}{15467}a-\frac{25806}{15467}$, $\frac{13}{15467}a^{12}-\frac{468}{15467}a^{10}+\frac{6318}{15467}a^{8}-\frac{160}{15467}a^{7}-\frac{39312}{15467}a^{6}+\frac{3360}{15467}a^{5}+\frac{110565}{15467}a^{4}-\frac{20160}{15467}a^{3}-\frac{113724}{15467}a^{2}+\frac{30240}{15467}a+\frac{3487}{15467}$, $\frac{1}{15467}a^{17}-\frac{51}{15467}a^{15}+\frac{1071}{15467}a^{13}-\frac{11934}{15467}a^{11}+\frac{75735}{15467}a^{9}-\frac{272646}{15467}a^{7}+\frac{520506}{15467}a^{5}-\frac{446148}{15467}a^{3}-\frac{1801}{15467}a^{2}+\frac{111537}{15467}a+\frac{10806}{15467}$, $\frac{1}{15467}a^{17}-\frac{51}{15467}a^{15}+\frac{1074}{15467}a^{13}-\frac{12051}{15467}a^{11}-\frac{74}{15467}a^{10}+\frac{77459}{15467}a^{9}+\frac{2220}{15467}a^{8}-\frac{284445}{15467}a^{7}-\frac{22231}{15467}a^{6}+\frac{557199}{15467}a^{5}+\frac{80478}{15467}a^{4}-\frac{487377}{15467}a^{3}-\frac{64252}{15467}a^{2}+\frac{117369}{15467}a-\frac{11496}{15467}$, $\frac{1}{15467}a^{17}-\frac{51}{15467}a^{15}+\frac{15}{15467}a^{14}+\frac{1071}{15467}a^{13}-\frac{569}{15467}a^{12}-\frac{11934}{15467}a^{11}+\frac{8199}{15467}a^{10}+\frac{75735}{15467}a^{9}-\frac{55404}{15467}a^{8}-\frac{272207}{15467}a^{7}+\frac{172746}{15467}a^{6}+\frac{512005}{15467}a^{5}-\frac{195615}{15467}a^{4}-\frac{401604}{15467}a^{3}+\frac{386}{15467}a^{2}+\frac{60876}{15467}a+\frac{3200}{15467}$, $\frac{2}{15467}a^{16}-\frac{97}{15467}a^{14}+\frac{13}{15467}a^{13}+\frac{1901}{15467}a^{12}-\frac{542}{15467}a^{11}-\frac{19233}{15467}a^{10}+\frac{8760}{15467}a^{9}+\frac{106019}{15467}a^{8}-\frac{68456}{15467}a^{7}-\frac{305502}{15467}a^{6}+\frac{259972}{15467}a^{5}+\frac{390051}{15467}a^{4}-\frac{403094}{15467}a^{3}-\frac{131517}{15467}a^{2}+\frac{102366}{15467}a-\frac{3402}{15467}$, $\frac{35}{15467}a^{11}+\frac{179}{15467}a^{10}-\frac{871}{15467}a^{9}-\frac{5117}{15467}a^{8}+\frac{6192}{15467}a^{7}+\frac{50313}{15467}a^{6}-\frac{3753}{15467}a^{5}-\frac{196110}{15467}a^{4}-\frac{74115}{15467}a^{3}+\frac{253179}{15467}a^{2}+\frac{128948}{15467}a-\frac{15074}{15467}$, $\frac{1}{15467}a^{17}+\frac{2}{15467}a^{16}-\frac{56}{15467}a^{15}-\frac{112}{15467}a^{14}+\frac{1296}{15467}a^{13}+\frac{2544}{15467}a^{12}-\frac{16019}{15467}a^{11}-\frac{30096}{15467}a^{10}+\frac{114015}{15467}a^{9}+\frac{197387}{15467}a^{8}-\frac{468756}{15467}a^{7}-\frac{701544}{15467}a^{6}+\frac{1050744}{15467}a^{5}+\frac{1205678}{15467}a^{4}-\frac{1081463}{15467}a^{3}-\frac{735361}{15467}a^{2}+\frac{252657}{15467}a+\frac{68591}{15467}$, $\frac{1}{15467}a^{15}-\frac{1}{15467}a^{14}-\frac{29}{15467}a^{13}+\frac{29}{15467}a^{12}+\frac{186}{15467}a^{11}-\frac{151}{15467}a^{10}+\frac{1966}{15467}a^{9}-\frac{2868}{15467}a^{8}-\frac{31619}{15467}a^{7}+\frac{39407}{15467}a^{6}+\frac{147112}{15467}a^{5}-\frac{170382}{15467}a^{4}-\frac{240513}{15467}a^{3}+\frac{237528}{15467}a^{2}+\frac{62631}{15467}a-\frac{9157}{15467}$, $\frac{1}{15467}a^{17}+\frac{2}{15467}a^{16}-\frac{56}{15467}a^{15}-\frac{96}{15467}a^{14}+\frac{1312}{15467}a^{13}+\frac{1872}{15467}a^{12}-\frac{16682}{15467}a^{11}-\frac{19008}{15467}a^{10}+\frac{124662}{15467}a^{9}+\frac{106827}{15467}a^{8}-\frac{551592}{15467}a^{7}-\frac{323761}{15467}a^{6}+\frac{1369494}{15467}a^{5}+\frac{461648}{15467}a^{4}-\frac{1635971}{15467}a^{3}-\frac{184426}{15467}a^{2}+\frac{589401}{15467}a-\frac{23286}{15467}$, $\frac{5}{15467}a^{15}-\frac{17}{15467}a^{14}-\frac{225}{15467}a^{13}+\frac{714}{15467}a^{12}+\frac{4050}{15467}a^{11}-\frac{11750}{15467}a^{10}-\frac{36903}{15467}a^{9}+\frac{95460}{15467}a^{8}+\frac{176256}{15467}a^{7}-\frac{395073}{15467}a^{6}-\frac{408200}{15467}a^{5}+\frac{768544}{15467}a^{4}+\frac{373620}{15467}a^{3}-\frac{553098}{15467}a^{2}-\frac{131607}{15467}a+\frac{103150}{15467}$, $\frac{3}{15467}a^{17}+\frac{4}{15467}a^{16}-\frac{153}{15467}a^{15}-\frac{208}{15467}a^{14}+\frac{3197}{15467}a^{13}+\frac{4416}{15467}a^{12}-\frac{35104}{15467}a^{11}-\frac{49030}{15467}a^{10}+\frac{215434}{15467}a^{9}+\frac{302433}{15467}a^{8}-\frac{722079}{15467}a^{7}-\frac{1013729}{15467}a^{6}+\frac{1177536}{15467}a^{5}+\frac{1669446}{15467}a^{4}-\frac{645211}{15467}a^{3}-\frac{1060189}{15467}a^{2}-\frac{144279}{15467}a+\frac{42602}{15467}$ 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:  \( 3961676166.05 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 3961676166.05 \cdot 1}{2\cdot\sqrt{12922465537100419689226617716849}}\cr\approx \mathstrut & 0.144449554805 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 56*x^16 + 56*x^15 + 1312*x^14 - 1312*x^13 - 16643*x^12 + 16643*x^11 + 123406*x^10 - 123406*x^9 - 536825*x^8 + 536825*x^7 + 1291507*x^6 - 1291507*x^5 - 1450991*x^4 + 1450991*x^3 + 418894*x^2 - 418894*x + 44917)
 
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 - x^17 - 56*x^16 + 56*x^15 + 1312*x^14 - 1312*x^13 - 16643*x^12 + 16643*x^11 + 123406*x^10 - 123406*x^9 - 536825*x^8 + 536825*x^7 + 1291507*x^6 - 1291507*x^5 - 1450991*x^4 + 1450991*x^3 + 418894*x^2 - 418894*x + 44917, 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 - x^17 - 56*x^16 + 56*x^15 + 1312*x^14 - 1312*x^13 - 16643*x^12 + 16643*x^11 + 123406*x^10 - 123406*x^9 - 536825*x^8 + 536825*x^7 + 1291507*x^6 - 1291507*x^5 - 1450991*x^4 + 1450991*x^3 + 418894*x^2 - 418894*x + 44917);
 
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 - x^17 - 56*x^16 + 56*x^15 + 1312*x^14 - 1312*x^13 - 16643*x^12 + 16643*x^11 + 123406*x^10 - 123406*x^9 - 536825*x^8 + 536825*x^7 + 1291507*x^6 - 1291507*x^5 - 1450991*x^4 + 1450991*x^3 + 418894*x^2 - 418894*x + 44917);
 
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

$C_{18}$ (as 18T1):

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 cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{209}) \), 3.3.361.1, 6.6.3295687769.1, \(\Q(\zeta_{19})^+\)

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.9.0.1}{9} }^{2}$ $18$ ${\href{/padicField/5.9.0.1}{9} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{3}$ R ${\href{/padicField/13.9.0.1}{9} }^{2}$ $18$ R ${\href{/padicField/23.9.0.1}{9} }^{2}$ ${\href{/padicField/29.9.0.1}{9} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{3}$ ${\href{/padicField/37.2.0.1}{2} }^{9}$ ${\href{/padicField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/padicField/47.9.0.1}{9} }^{2}$ $18$ $18$

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
\(11\) Copy content Toggle raw display 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}$
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}$
\(19\) Copy content Toggle raw display 19.18.17.1$x^{18} + 342$$18$$1$$17$$C_{18}$$[\ ]_{18}$