Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 17*x^16 - 6*x^15 + 201*x^14 - 76*x^13 + 999*x^12 - 218*x^11 + 3519*x^10 - 623*x^9 + 5540*x^8 + 1505*x^7 + 5069*x^6 - 129*x^5 + 528*x^4 - 97*x^3 + 56*x^2 - 7*x + 1)
gp: K = bnfinit(y^18 - y^17 + 17*y^16 - 6*y^15 + 201*y^14 - 76*y^13 + 999*y^12 - 218*y^11 + 3519*y^10 - 623*y^9 + 5540*y^8 + 1505*y^7 + 5069*y^6 - 129*y^5 + 528*y^4 - 97*y^3 + 56*y^2 - 7*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 + 17*x^16 - 6*x^15 + 201*x^14 - 76*x^13 + 999*x^12 - 218*x^11 + 3519*x^10 - 623*x^9 + 5540*x^8 + 1505*x^7 + 5069*x^6 - 129*x^5 + 528*x^4 - 97*x^3 + 56*x^2 - 7*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - x^17 + 17*x^16 - 6*x^15 + 201*x^14 - 76*x^13 + 999*x^12 - 218*x^11 + 3519*x^10 - 623*x^9 + 5540*x^8 + 1505*x^7 + 5069*x^6 - 129*x^5 + 528*x^4 - 97*x^3 + 56*x^2 - 7*x + 1)
\( x^{18} - x^{17} + 17 x^{16} - 6 x^{15} + 201 x^{14} - 76 x^{13} + 999 x^{12} - 218 x^{11} + 3519 x^{10} + \cdots + 1 \)
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: |
\(-242839247007536485508643885603\)
\(\medspace = -\,3^{9}\cdot 37^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(42.91\) | 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: | $3^{1/2}37^{8/9}\approx 42.90596907675813$ | ||
| Ramified primes: |
\(3\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(111=3\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{111}(1,·)$, $\chi_{111}(70,·)$, $\chi_{111}(7,·)$, $\chi_{111}(10,·)$, $\chi_{111}(16,·)$, $\chi_{111}(83,·)$, $\chi_{111}(86,·)$, $\chi_{111}(71,·)$, $\chi_{111}(26,·)$, $\chi_{111}(34,·)$, $\chi_{111}(100,·)$, $\chi_{111}(38,·)$, $\chi_{111}(107,·)$, $\chi_{111}(44,·)$, $\chi_{111}(46,·)$, $\chi_{111}(47,·)$, $\chi_{111}(49,·)$, $\chi_{111}(53,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{256}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{31}a^{14}+\frac{12}{31}a^{13}+\frac{3}{31}a^{12}+\frac{10}{31}a^{11}-\frac{1}{31}a^{10}+\frac{3}{31}a^{9}-\frac{15}{31}a^{8}-\frac{5}{31}a^{7}-\frac{3}{31}a^{6}+\frac{10}{31}a^{4}-\frac{9}{31}a^{3}-\frac{5}{31}a+\frac{1}{31}$, $\frac{1}{1333}a^{15}-\frac{5}{1333}a^{14}+\frac{605}{1333}a^{13}-\frac{227}{1333}a^{12}+\frac{666}{1333}a^{11}-\frac{631}{1333}a^{10}-\frac{469}{1333}a^{9}-\frac{370}{1333}a^{8}-\frac{228}{1333}a^{7}-\frac{42}{1333}a^{6}-\frac{207}{1333}a^{5}+\frac{131}{1333}a^{4}+\frac{153}{1333}a^{3}-\frac{315}{1333}a^{2}+\frac{644}{1333}a+\frac{231}{1333}$, $\frac{1}{1333}a^{16}+\frac{21}{1333}a^{14}+\frac{89}{1333}a^{13}+\frac{520}{1333}a^{12}-\frac{225}{1333}a^{11}-\frac{399}{1333}a^{10}-\frac{393}{1333}a^{9}-\frac{358}{1333}a^{8}+\frac{280}{1333}a^{7}-\frac{73}{1333}a^{6}+\frac{429}{1333}a^{5}+\frac{550}{1333}a^{4}+\frac{149}{1333}a^{3}+\frac{402}{1333}a^{2}-\frac{419}{1333}a+\frac{596}{1333}$, $\frac{1}{14\!\cdots\!21}a^{17}+\frac{26\!\cdots\!32}{14\!\cdots\!21}a^{16}+\frac{20\!\cdots\!12}{14\!\cdots\!21}a^{15}-\frac{11\!\cdots\!82}{14\!\cdots\!21}a^{14}+\frac{92\!\cdots\!87}{14\!\cdots\!21}a^{13}+\frac{54\!\cdots\!63}{14\!\cdots\!21}a^{12}+\frac{12\!\cdots\!79}{14\!\cdots\!21}a^{11}+\frac{47\!\cdots\!66}{14\!\cdots\!21}a^{10}-\frac{44\!\cdots\!09}{14\!\cdots\!21}a^{9}-\frac{35\!\cdots\!65}{14\!\cdots\!21}a^{8}-\frac{66\!\cdots\!92}{14\!\cdots\!21}a^{7}-\frac{50\!\cdots\!66}{14\!\cdots\!21}a^{6}+\frac{14\!\cdots\!30}{14\!\cdots\!21}a^{5}-\frac{74\!\cdots\!58}{14\!\cdots\!21}a^{4}-\frac{19\!\cdots\!74}{14\!\cdots\!21}a^{3}+\frac{21\!\cdots\!20}{14\!\cdots\!21}a^{2}-\frac{64\!\cdots\!20}{14\!\cdots\!21}a+\frac{68\!\cdots\!45}{14\!\cdots\!21}$
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
$C_{171}$, which has order $171$ (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: |
\( \frac{20025969607191039788}{145926445271948161921} a^{17} - \frac{20165950282932610000}{145926445271948161921} a^{16} + \frac{340315133056582183978}{145926445271948161921} a^{15} - \frac{121988076748962363963}{145926445271948161921} a^{14} + \frac{4021290298088786965742}{145926445271948161921} a^{13} - \frac{1543805908980430236049}{145926445271948161921} a^{12} + \frac{19962041145841092640272}{145926445271948161921} a^{11} - \frac{4429506202037532132371}{145926445271948161921} a^{10} + \frac{70222182484462262383690}{145926445271948161921} a^{9} - \frac{407706932425405162452}{4707304686191876191} a^{8} + \frac{110070180432231351495904}{145926445271948161921} a^{7} + \frac{30485040438604793540912}{145926445271948161921} a^{6} + \frac{99776704753695173021893}{145926445271948161921} a^{5} - \frac{2246476066609438087994}{145926445271948161921} a^{4} + \frac{9841717502401981062185}{145926445271948161921} a^{3} - \frac{550607737877098968877}{145926445271948161921} a^{2} + \frac{33525759933045100845}{4707304686191876191} a + \frac{16237411536556207962}{145926445271948161921} \)
(order $6$)
| 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: |
$\frac{23\!\cdots\!45}{14\!\cdots\!21}a^{17}-\frac{38\!\cdots\!85}{14\!\cdots\!21}a^{16}+\frac{41\!\cdots\!50}{14\!\cdots\!21}a^{15}-\frac{39\!\cdots\!57}{14\!\cdots\!21}a^{14}+\frac{47\!\cdots\!40}{14\!\cdots\!21}a^{13}-\frac{48\!\cdots\!98}{14\!\cdots\!21}a^{12}+\frac{54\!\cdots\!54}{33\!\cdots\!47}a^{11}-\frac{20\!\cdots\!95}{14\!\cdots\!21}a^{10}+\frac{78\!\cdots\!56}{14\!\cdots\!21}a^{9}-\frac{68\!\cdots\!84}{14\!\cdots\!21}a^{8}+\frac{11\!\cdots\!03}{14\!\cdots\!21}a^{7}-\frac{49\!\cdots\!07}{14\!\cdots\!21}a^{6}+\frac{51\!\cdots\!45}{14\!\cdots\!21}a^{5}-\frac{98\!\cdots\!40}{14\!\cdots\!21}a^{4}-\frac{34\!\cdots\!11}{14\!\cdots\!21}a^{3}-\frac{11\!\cdots\!56}{14\!\cdots\!21}a^{2}+\frac{14\!\cdots\!31}{14\!\cdots\!21}a+\frac{24\!\cdots\!56}{14\!\cdots\!21}$, $a$, $\frac{11\!\cdots\!79}{14\!\cdots\!21}a^{17}-\frac{21\!\cdots\!39}{14\!\cdots\!21}a^{16}+\frac{20\!\cdots\!10}{14\!\cdots\!21}a^{15}-\frac{22\!\cdots\!63}{14\!\cdots\!21}a^{14}+\frac{23\!\cdots\!22}{14\!\cdots\!21}a^{13}-\frac{27\!\cdots\!30}{14\!\cdots\!21}a^{12}+\frac{11\!\cdots\!84}{14\!\cdots\!21}a^{11}-\frac{11\!\cdots\!81}{14\!\cdots\!21}a^{10}+\frac{40\!\cdots\!01}{14\!\cdots\!21}a^{9}-\frac{12\!\cdots\!28}{47\!\cdots\!91}a^{8}+\frac{60\!\cdots\!81}{14\!\cdots\!21}a^{7}-\frac{32\!\cdots\!89}{14\!\cdots\!21}a^{6}+\frac{91\!\cdots\!29}{47\!\cdots\!91}a^{5}-\frac{54\!\cdots\!82}{14\!\cdots\!21}a^{4}-\frac{92\!\cdots\!14}{14\!\cdots\!21}a^{3}-\frac{61\!\cdots\!34}{14\!\cdots\!21}a^{2}+\frac{78\!\cdots\!11}{14\!\cdots\!21}a-\frac{26\!\cdots\!14}{14\!\cdots\!21}$, $\frac{15\!\cdots\!33}{14\!\cdots\!21}a^{17}-\frac{62\!\cdots\!08}{14\!\cdots\!21}a^{16}+\frac{32\!\cdots\!17}{14\!\cdots\!21}a^{15}-\frac{91\!\cdots\!66}{14\!\cdots\!21}a^{14}+\frac{36\!\cdots\!09}{14\!\cdots\!21}a^{13}-\frac{10\!\cdots\!29}{14\!\cdots\!21}a^{12}+\frac{22\!\cdots\!23}{14\!\cdots\!21}a^{11}-\frac{51\!\cdots\!87}{14\!\cdots\!21}a^{10}+\frac{82\!\cdots\!98}{14\!\cdots\!21}a^{9}-\frac{17\!\cdots\!16}{14\!\cdots\!21}a^{8}+\frac{17\!\cdots\!22}{14\!\cdots\!21}a^{7}-\frac{24\!\cdots\!27}{14\!\cdots\!21}a^{6}+\frac{99\!\cdots\!38}{14\!\cdots\!21}a^{5}-\frac{20\!\cdots\!14}{14\!\cdots\!21}a^{4}+\frac{96\!\cdots\!05}{14\!\cdots\!21}a^{3}-\frac{22\!\cdots\!53}{14\!\cdots\!21}a^{2}+\frac{28\!\cdots\!72}{14\!\cdots\!21}a-\frac{67\!\cdots\!02}{14\!\cdots\!21}$, $\frac{56\!\cdots\!88}{14\!\cdots\!21}a^{17}-\frac{55\!\cdots\!93}{14\!\cdots\!21}a^{16}+\frac{96\!\cdots\!55}{14\!\cdots\!21}a^{15}-\frac{30\!\cdots\!35}{14\!\cdots\!21}a^{14}+\frac{11\!\cdots\!99}{14\!\cdots\!21}a^{13}-\frac{39\!\cdots\!85}{14\!\cdots\!21}a^{12}+\frac{56\!\cdots\!25}{14\!\cdots\!21}a^{11}-\frac{10\!\cdots\!37}{14\!\cdots\!21}a^{10}+\frac{19\!\cdots\!37}{14\!\cdots\!21}a^{9}-\frac{28\!\cdots\!00}{14\!\cdots\!21}a^{8}+\frac{99\!\cdots\!32}{47\!\cdots\!91}a^{7}+\frac{95\!\cdots\!81}{14\!\cdots\!21}a^{6}+\frac{28\!\cdots\!47}{14\!\cdots\!21}a^{5}-\frac{22\!\cdots\!99}{14\!\cdots\!21}a^{4}+\frac{20\!\cdots\!09}{14\!\cdots\!21}a^{3}-\frac{60\!\cdots\!32}{14\!\cdots\!21}a^{2}+\frac{20\!\cdots\!03}{14\!\cdots\!21}a-\frac{25\!\cdots\!58}{14\!\cdots\!21}$, $\frac{37\!\cdots\!39}{14\!\cdots\!21}a^{17}-\frac{35\!\cdots\!35}{14\!\cdots\!21}a^{16}+\frac{20\!\cdots\!27}{47\!\cdots\!91}a^{15}-\frac{18\!\cdots\!84}{14\!\cdots\!21}a^{14}+\frac{73\!\cdots\!97}{14\!\cdots\!21}a^{13}-\frac{24\!\cdots\!41}{14\!\cdots\!21}a^{12}+\frac{36\!\cdots\!77}{14\!\cdots\!21}a^{11}-\frac{57\!\cdots\!53}{14\!\cdots\!21}a^{10}+\frac{12\!\cdots\!55}{14\!\cdots\!21}a^{9}-\frac{14\!\cdots\!10}{14\!\cdots\!21}a^{8}+\frac{19\!\cdots\!24}{14\!\cdots\!21}a^{7}+\frac{70\!\cdots\!87}{14\!\cdots\!21}a^{6}+\frac{17\!\cdots\!05}{14\!\cdots\!21}a^{5}+\frac{32\!\cdots\!59}{14\!\cdots\!21}a^{4}+\frac{82\!\cdots\!35}{33\!\cdots\!47}a^{3}+\frac{66\!\cdots\!85}{14\!\cdots\!21}a^{2}+\frac{28\!\cdots\!67}{14\!\cdots\!21}a-\frac{30\!\cdots\!88}{14\!\cdots\!21}$, $\frac{67\!\cdots\!54}{14\!\cdots\!21}a^{17}-\frac{57\!\cdots\!21}{14\!\cdots\!21}a^{16}+\frac{11\!\cdots\!08}{14\!\cdots\!21}a^{15}-\frac{24\!\cdots\!46}{14\!\cdots\!21}a^{14}+\frac{13\!\cdots\!20}{14\!\cdots\!21}a^{13}-\frac{31\!\cdots\!62}{14\!\cdots\!21}a^{12}+\frac{66\!\cdots\!22}{14\!\cdots\!21}a^{11}-\frac{50\!\cdots\!40}{14\!\cdots\!21}a^{10}+\frac{23\!\cdots\!70}{14\!\cdots\!21}a^{9}-\frac{78\!\cdots\!12}{14\!\cdots\!21}a^{8}+\frac{36\!\cdots\!46}{14\!\cdots\!21}a^{7}+\frac{15\!\cdots\!54}{14\!\cdots\!21}a^{6}+\frac{35\!\cdots\!28}{14\!\cdots\!21}a^{5}+\frac{41\!\cdots\!28}{14\!\cdots\!21}a^{4}+\frac{34\!\cdots\!38}{14\!\cdots\!21}a^{3}-\frac{14\!\cdots\!98}{14\!\cdots\!21}a^{2}+\frac{23\!\cdots\!40}{14\!\cdots\!21}a+\frac{67\!\cdots\!62}{14\!\cdots\!21}$, $\frac{72\!\cdots\!18}{14\!\cdots\!21}a^{17}-\frac{70\!\cdots\!46}{14\!\cdots\!21}a^{16}+\frac{12\!\cdots\!14}{14\!\cdots\!21}a^{15}-\frac{40\!\cdots\!18}{14\!\cdots\!21}a^{14}+\frac{14\!\cdots\!46}{14\!\cdots\!21}a^{13}-\frac{51\!\cdots\!12}{14\!\cdots\!21}a^{12}+\frac{72\!\cdots\!50}{14\!\cdots\!21}a^{11}-\frac{14\!\cdots\!44}{14\!\cdots\!21}a^{10}+\frac{25\!\cdots\!74}{14\!\cdots\!21}a^{9}-\frac{39\!\cdots\!27}{14\!\cdots\!21}a^{8}+\frac{39\!\cdots\!80}{14\!\cdots\!21}a^{7}+\frac{11\!\cdots\!70}{14\!\cdots\!21}a^{6}+\frac{36\!\cdots\!44}{14\!\cdots\!21}a^{5}-\frac{57\!\cdots\!50}{14\!\cdots\!21}a^{4}+\frac{31\!\cdots\!08}{14\!\cdots\!21}a^{3}-\frac{93\!\cdots\!84}{14\!\cdots\!21}a^{2}+\frac{32\!\cdots\!52}{14\!\cdots\!21}a-\frac{40\!\cdots\!34}{14\!\cdots\!21}$
| 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: | \( 409151.310213 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 409151.310213 \cdot 171}{6\cdot\sqrt{242839247007536485508643885603}}\cr\approx \mathstrut & 0.361150468598 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 17*x^16 - 6*x^15 + 201*x^14 - 76*x^13 + 999*x^12 - 218*x^11 + 3519*x^10 - 623*x^9 + 5540*x^8 + 1505*x^7 + 5069*x^6 - 129*x^5 + 528*x^4 - 97*x^3 + 56*x^2 - 7*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 - x^17 + 17*x^16 - 6*x^15 + 201*x^14 - 76*x^13 + 999*x^12 - 218*x^11 + 3519*x^10 - 623*x^9 + 5540*x^8 + 1505*x^7 + 5069*x^6 - 129*x^5 + 528*x^4 - 97*x^3 + 56*x^2 - 7*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 - x^17 + 17*x^16 - 6*x^15 + 201*x^14 - 76*x^13 + 999*x^12 - 218*x^11 + 3519*x^10 - 623*x^9 + 5540*x^8 + 1505*x^7 + 5069*x^6 - 129*x^5 + 528*x^4 - 97*x^3 + 56*x^2 - 7*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 - x^17 + 17*x^16 - 6*x^15 + 201*x^14 - 76*x^13 + 999*x^12 - 218*x^11 + 3519*x^10 - 623*x^9 + 5540*x^8 + 1505*x^7 + 5069*x^6 - 129*x^5 + 528*x^4 - 97*x^3 + 56*x^2 - 7*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
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
| \(\Q(\sqrt{-3}) \), 3.3.1369.1, 6.0.50602347.1, 9.9.3512479453921.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $18$ | R | $18$ | ${\href{/padicField/7.9.0.1}{9} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/19.9.0.1}{9} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.1.0.1}{1} }^{18}$ | R | $18$ | ${\href{/padicField/43.1.0.1}{1} }^{18}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | $18$ | $18$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $18$ | $2$ | $9$ | $9$ | |||
|
\(37\)
| 37.9.8.1 | $x^{9} + 37$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 37.9.8.1 | $x^{9} + 37$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |