Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 2*x^16 - 26*x^15 + 42*x^14 + 90*x^13 + 431*x^12 - 978*x^11 - 1417*x^10 - 232*x^9 + 7265*x^8 + 2468*x^7 - 13003*x^6 - 10038*x^5 + 39442*x^4 - 58082*x^3 + 97785*x^2 - 127756*x + 68921)
gp: K = bnfinit(y^18 - 2*y^17 - 2*y^16 - 26*y^15 + 42*y^14 + 90*y^13 + 431*y^12 - 978*y^11 - 1417*y^10 - 232*y^9 + 7265*y^8 + 2468*y^7 - 13003*y^6 - 10038*y^5 + 39442*y^4 - 58082*y^3 + 97785*y^2 - 127756*y + 68921, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 - 2*x^16 - 26*x^15 + 42*x^14 + 90*x^13 + 431*x^12 - 978*x^11 - 1417*x^10 - 232*x^9 + 7265*x^8 + 2468*x^7 - 13003*x^6 - 10038*x^5 + 39442*x^4 - 58082*x^3 + 97785*x^2 - 127756*x + 68921);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 2*x^17 - 2*x^16 - 26*x^15 + 42*x^14 + 90*x^13 + 431*x^12 - 978*x^11 - 1417*x^10 - 232*x^9 + 7265*x^8 + 2468*x^7 - 13003*x^6 - 10038*x^5 + 39442*x^4 - 58082*x^3 + 97785*x^2 - 127756*x + 68921)
\( x^{18} - 2 x^{17} - 2 x^{16} - 26 x^{15} + 42 x^{14} + 90 x^{13} + 431 x^{12} - 978 x^{11} + \cdots + 68921 \)
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: |
\(-4641657809149437673472000000000\)
\(\medspace = -\,2^{30}\cdot 5^{9}\cdot 19^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(50.55\) | 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: | $2^{2}5^{1/2}19^{2/3}\approx 63.686501757102$ | ||
| Ramified primes: |
\(2\), \(5\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{41}a^{16}-\frac{2}{41}a^{15}-\frac{2}{41}a^{14}+\frac{15}{41}a^{13}+\frac{1}{41}a^{12}+\frac{8}{41}a^{11}-\frac{20}{41}a^{10}+\frac{6}{41}a^{9}+\frac{18}{41}a^{8}+\frac{14}{41}a^{7}+\frac{8}{41}a^{6}+\frac{8}{41}a^{5}-\frac{6}{41}a^{4}+\frac{7}{41}a^{3}+\frac{15}{41}a$, $\frac{1}{89\!\cdots\!23}a^{17}+\frac{83\!\cdots\!10}{89\!\cdots\!23}a^{16}+\frac{19\!\cdots\!43}{89\!\cdots\!23}a^{15}-\frac{38\!\cdots\!28}{89\!\cdots\!23}a^{14}-\frac{12\!\cdots\!74}{89\!\cdots\!23}a^{13}-\frac{19\!\cdots\!31}{89\!\cdots\!23}a^{12}+\frac{34\!\cdots\!46}{89\!\cdots\!23}a^{11}+\frac{19\!\cdots\!08}{89\!\cdots\!23}a^{10}-\frac{36\!\cdots\!03}{89\!\cdots\!23}a^{9}+\frac{36\!\cdots\!51}{89\!\cdots\!23}a^{8}+\frac{23\!\cdots\!59}{89\!\cdots\!23}a^{7}-\frac{29\!\cdots\!54}{89\!\cdots\!23}a^{6}-\frac{54\!\cdots\!10}{19\!\cdots\!69}a^{5}+\frac{53\!\cdots\!50}{89\!\cdots\!23}a^{4}-\frac{58\!\cdots\!65}{21\!\cdots\!03}a^{3}+\frac{36\!\cdots\!27}{89\!\cdots\!23}a^{2}+\frac{71\!\cdots\!77}{21\!\cdots\!03}a+\frac{10\!\cdots\!10}{53\!\cdots\!83}$
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
$C_{14}$, which has order $14$ (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 $2$)
| 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: |
$\frac{24\!\cdots\!71}{89\!\cdots\!23}a^{17}-\frac{18\!\cdots\!96}{89\!\cdots\!23}a^{16}-\frac{74\!\cdots\!24}{89\!\cdots\!23}a^{15}-\frac{73\!\cdots\!03}{89\!\cdots\!23}a^{14}+\frac{12\!\cdots\!48}{89\!\cdots\!23}a^{13}+\frac{24\!\cdots\!29}{89\!\cdots\!23}a^{12}+\frac{13\!\cdots\!42}{89\!\cdots\!23}a^{11}-\frac{71\!\cdots\!94}{89\!\cdots\!23}a^{10}-\frac{44\!\cdots\!60}{89\!\cdots\!23}a^{9}-\frac{61\!\cdots\!32}{89\!\cdots\!23}a^{8}+\frac{10\!\cdots\!46}{89\!\cdots\!23}a^{7}+\frac{19\!\cdots\!77}{89\!\cdots\!23}a^{6}-\frac{17\!\cdots\!22}{19\!\cdots\!69}a^{5}-\frac{35\!\cdots\!69}{89\!\cdots\!23}a^{4}+\frac{13\!\cdots\!95}{21\!\cdots\!03}a^{3}-\frac{76\!\cdots\!34}{89\!\cdots\!23}a^{2}+\frac{35\!\cdots\!88}{21\!\cdots\!03}a-\frac{79\!\cdots\!28}{53\!\cdots\!83}$, $\frac{26\!\cdots\!58}{89\!\cdots\!23}a^{17}-\frac{19\!\cdots\!58}{89\!\cdots\!23}a^{16}-\frac{62\!\cdots\!33}{89\!\cdots\!23}a^{15}-\frac{75\!\cdots\!69}{89\!\cdots\!23}a^{14}+\frac{71\!\cdots\!41}{89\!\cdots\!23}a^{13}+\frac{19\!\cdots\!31}{89\!\cdots\!23}a^{12}+\frac{13\!\cdots\!43}{89\!\cdots\!23}a^{11}-\frac{64\!\cdots\!15}{89\!\cdots\!23}a^{10}-\frac{34\!\cdots\!35}{89\!\cdots\!23}a^{9}-\frac{49\!\cdots\!53}{89\!\cdots\!23}a^{8}+\frac{85\!\cdots\!89}{89\!\cdots\!23}a^{7}+\frac{99\!\cdots\!39}{89\!\cdots\!23}a^{6}-\frac{39\!\cdots\!03}{19\!\cdots\!69}a^{5}-\frac{22\!\cdots\!27}{89\!\cdots\!23}a^{4}+\frac{22\!\cdots\!09}{21\!\cdots\!03}a^{3}-\frac{56\!\cdots\!22}{89\!\cdots\!23}a^{2}+\frac{41\!\cdots\!36}{21\!\cdots\!03}a-\frac{96\!\cdots\!43}{53\!\cdots\!83}$, $\frac{73\!\cdots\!88}{89\!\cdots\!23}a^{17}-\frac{41\!\cdots\!16}{89\!\cdots\!23}a^{16}-\frac{21\!\cdots\!77}{89\!\cdots\!23}a^{15}-\frac{22\!\cdots\!78}{89\!\cdots\!23}a^{14}-\frac{11\!\cdots\!03}{89\!\cdots\!23}a^{13}+\frac{69\!\cdots\!00}{89\!\cdots\!23}a^{12}+\frac{43\!\cdots\!89}{89\!\cdots\!23}a^{11}-\frac{10\!\cdots\!36}{89\!\cdots\!23}a^{10}-\frac{13\!\cdots\!89}{89\!\cdots\!23}a^{9}-\frac{23\!\cdots\!26}{89\!\cdots\!23}a^{8}+\frac{23\!\cdots\!57}{89\!\cdots\!23}a^{7}+\frac{65\!\cdots\!39}{89\!\cdots\!23}a^{6}+\frac{55\!\cdots\!51}{19\!\cdots\!69}a^{5}-\frac{10\!\cdots\!93}{89\!\cdots\!23}a^{4}+\frac{29\!\cdots\!35}{21\!\cdots\!03}a^{3}-\frac{23\!\cdots\!35}{89\!\cdots\!23}a^{2}+\frac{93\!\cdots\!48}{21\!\cdots\!03}a-\frac{23\!\cdots\!60}{53\!\cdots\!83}$, $\frac{75\!\cdots\!06}{89\!\cdots\!23}a^{17}+\frac{22\!\cdots\!69}{89\!\cdots\!23}a^{16}-\frac{38\!\cdots\!92}{89\!\cdots\!23}a^{15}-\frac{26\!\cdots\!20}{89\!\cdots\!23}a^{14}-\frac{18\!\cdots\!59}{89\!\cdots\!23}a^{13}+\frac{11\!\cdots\!42}{89\!\cdots\!23}a^{12}+\frac{56\!\cdots\!07}{89\!\cdots\!23}a^{11}+\frac{23\!\cdots\!50}{89\!\cdots\!23}a^{10}-\frac{23\!\cdots\!87}{89\!\cdots\!23}a^{9}-\frac{46\!\cdots\!20}{89\!\cdots\!23}a^{8}+\frac{93\!\cdots\!21}{89\!\cdots\!23}a^{7}+\frac{16\!\cdots\!11}{89\!\cdots\!23}a^{6}+\frac{30\!\cdots\!71}{19\!\cdots\!69}a^{5}-\frac{84\!\cdots\!50}{89\!\cdots\!23}a^{4}-\frac{77\!\cdots\!23}{21\!\cdots\!03}a^{3}-\frac{40\!\cdots\!32}{89\!\cdots\!23}a^{2}-\frac{25\!\cdots\!39}{21\!\cdots\!03}a+\frac{44\!\cdots\!34}{53\!\cdots\!83}$, $\frac{59\!\cdots\!26}{89\!\cdots\!23}a^{17}+\frac{59\!\cdots\!59}{89\!\cdots\!23}a^{16}-\frac{52\!\cdots\!38}{89\!\cdots\!23}a^{15}-\frac{17\!\cdots\!84}{89\!\cdots\!23}a^{14}-\frac{29\!\cdots\!07}{89\!\cdots\!23}a^{13}-\frac{10\!\cdots\!26}{89\!\cdots\!23}a^{12}+\frac{29\!\cdots\!61}{89\!\cdots\!23}a^{11}+\frac{35\!\cdots\!45}{89\!\cdots\!23}a^{10}-\frac{27\!\cdots\!34}{89\!\cdots\!23}a^{9}-\frac{14\!\cdots\!21}{89\!\cdots\!23}a^{8}-\frac{26\!\cdots\!98}{89\!\cdots\!23}a^{7}+\frac{25\!\cdots\!08}{89\!\cdots\!23}a^{6}+\frac{24\!\cdots\!29}{19\!\cdots\!69}a^{5}-\frac{63\!\cdots\!68}{89\!\cdots\!23}a^{4}+\frac{19\!\cdots\!15}{21\!\cdots\!03}a^{3}-\frac{15\!\cdots\!96}{89\!\cdots\!23}a^{2}+\frac{54\!\cdots\!08}{21\!\cdots\!03}a-\frac{58\!\cdots\!57}{53\!\cdots\!83}$, $\frac{99\!\cdots\!76}{89\!\cdots\!23}a^{17}-\frac{14\!\cdots\!12}{89\!\cdots\!23}a^{16}-\frac{31\!\cdots\!42}{89\!\cdots\!23}a^{15}-\frac{27\!\cdots\!58}{89\!\cdots\!23}a^{14}+\frac{28\!\cdots\!89}{89\!\cdots\!23}a^{13}+\frac{11\!\cdots\!80}{89\!\cdots\!23}a^{12}+\frac{50\!\cdots\!39}{89\!\cdots\!23}a^{11}-\frac{77\!\cdots\!96}{89\!\cdots\!23}a^{10}-\frac{20\!\cdots\!87}{89\!\cdots\!23}a^{9}-\frac{14\!\cdots\!67}{89\!\cdots\!23}a^{8}+\frac{81\!\cdots\!91}{89\!\cdots\!23}a^{7}+\frac{83\!\cdots\!99}{89\!\cdots\!23}a^{6}-\frac{22\!\cdots\!93}{19\!\cdots\!69}a^{5}-\frac{27\!\cdots\!99}{89\!\cdots\!23}a^{4}+\frac{57\!\cdots\!24}{21\!\cdots\!03}a^{3}-\frac{27\!\cdots\!01}{89\!\cdots\!23}a^{2}+\frac{23\!\cdots\!19}{21\!\cdots\!03}a-\frac{42\!\cdots\!87}{53\!\cdots\!83}$, $\frac{67\!\cdots\!10}{89\!\cdots\!23}a^{17}-\frac{44\!\cdots\!76}{89\!\cdots\!23}a^{16}-\frac{19\!\cdots\!72}{89\!\cdots\!23}a^{15}-\frac{20\!\cdots\!21}{89\!\cdots\!23}a^{14}+\frac{11\!\cdots\!43}{89\!\cdots\!23}a^{13}+\frac{63\!\cdots\!72}{89\!\cdots\!23}a^{12}+\frac{37\!\cdots\!53}{89\!\cdots\!23}a^{11}-\frac{15\!\cdots\!27}{89\!\cdots\!23}a^{10}-\frac{11\!\cdots\!56}{89\!\cdots\!23}a^{9}-\frac{17\!\cdots\!57}{89\!\cdots\!23}a^{8}+\frac{25\!\cdots\!52}{89\!\cdots\!23}a^{7}+\frac{52\!\cdots\!49}{89\!\cdots\!23}a^{6}-\frac{37\!\cdots\!48}{19\!\cdots\!69}a^{5}-\frac{93\!\cdots\!63}{89\!\cdots\!23}a^{4}+\frac{34\!\cdots\!09}{21\!\cdots\!03}a^{3}-\frac{20\!\cdots\!03}{89\!\cdots\!23}a^{2}+\frac{94\!\cdots\!73}{21\!\cdots\!03}a-\frac{20\!\cdots\!43}{53\!\cdots\!83}$, $\frac{16\!\cdots\!83}{89\!\cdots\!23}a^{17}-\frac{90\!\cdots\!68}{89\!\cdots\!23}a^{16}-\frac{49\!\cdots\!31}{89\!\cdots\!23}a^{15}-\frac{51\!\cdots\!70}{89\!\cdots\!23}a^{14}-\frac{38\!\cdots\!51}{89\!\cdots\!23}a^{13}+\frac{15\!\cdots\!06}{89\!\cdots\!23}a^{12}+\frac{96\!\cdots\!53}{89\!\cdots\!23}a^{11}-\frac{23\!\cdots\!05}{89\!\cdots\!23}a^{10}-\frac{29\!\cdots\!67}{89\!\cdots\!23}a^{9}-\frac{48\!\cdots\!40}{89\!\cdots\!23}a^{8}+\frac{53\!\cdots\!79}{89\!\cdots\!23}a^{7}+\frac{13\!\cdots\!74}{89\!\cdots\!23}a^{6}-\frac{38\!\cdots\!53}{19\!\cdots\!69}a^{5}-\frac{21\!\cdots\!16}{89\!\cdots\!23}a^{4}+\frac{82\!\cdots\!02}{21\!\cdots\!03}a^{3}-\frac{48\!\cdots\!70}{89\!\cdots\!23}a^{2}+\frac{22\!\cdots\!34}{21\!\cdots\!03}a-\frac{44\!\cdots\!85}{53\!\cdots\!83}$
| 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: | \( 3468123.856372189 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 3468123.856372189 \cdot 14}{2\cdot\sqrt{4641657809149437673472000000000}}\cr\approx \mathstrut & 0.171978791964417 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 2*x^16 - 26*x^15 + 42*x^14 + 90*x^13 + 431*x^12 - 978*x^11 - 1417*x^10 - 232*x^9 + 7265*x^8 + 2468*x^7 - 13003*x^6 - 10038*x^5 + 39442*x^4 - 58082*x^3 + 97785*x^2 - 127756*x + 68921)
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 - 2*x^17 - 2*x^16 - 26*x^15 + 42*x^14 + 90*x^13 + 431*x^12 - 978*x^11 - 1417*x^10 - 232*x^9 + 7265*x^8 + 2468*x^7 - 13003*x^6 - 10038*x^5 + 39442*x^4 - 58082*x^3 + 97785*x^2 - 127756*x + 68921, 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 - 2*x^17 - 2*x^16 - 26*x^15 + 42*x^14 + 90*x^13 + 431*x^12 - 978*x^11 - 1417*x^10 - 232*x^9 + 7265*x^8 + 2468*x^7 - 13003*x^6 - 10038*x^5 + 39442*x^4 - 58082*x^3 + 97785*x^2 - 127756*x + 68921);
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 - 2*x^17 - 2*x^16 - 26*x^15 + 42*x^14 + 90*x^13 + 431*x^12 - 978*x^11 - 1417*x^10 - 232*x^9 + 7265*x^8 + 2468*x^7 - 13003*x^6 - 10038*x^5 + 39442*x^4 - 58082*x^3 + 97785*x^2 - 127756*x + 68921);
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
$C_6\times S_3$ (as 18T6):
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 18 conjugacy class representatives for $S_3 \times C_6$ |
| Character table for $S_3 \times C_6$ |
Intermediate fields
| \(\Q(\sqrt{-5}) \), 3.3.361.1, 3.1.14440.1, 6.0.1042568000.1, 6.0.16681088000.5, 9.3.3010936384000.1 |
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 12 sibling: | 12.0.34162868224000000.1 |
| Degree 18 sibling: | 18.6.297066099785564011102208000000.1 |
| Minimal sibling: | 12.0.34162868224000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{9}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.6.5 | $x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
| 2.12.24.318 | $x^{12} + 10 x^{10} + 12 x^{9} + 110 x^{8} + 80 x^{7} + 752 x^{6} + 512 x^{5} + 1636 x^{4} + 1504 x^{3} + 1224 x^{2} + 1008 x - 648$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $[2, 3]^{3}$ | |
|
\(5\)
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(19\)
| 19.18.12.1 | $x^{18} + 1938 x^{16} + 165 x^{15} + 1564986 x^{14} + 217074 x^{13} + 673956770 x^{12} + 124220751 x^{11} + 163260541175 x^{10} + 40698676942 x^{9} + 21101810097561 x^{8} + 7872565858164 x^{7} + 1139107203476720 x^{6} + 760256762812749 x^{5} + 441034593923007 x^{4} + 19443033036221259 x^{3} + 22753074450170540 x^{2} + 9283251966890258 x + 2743830934025437$ | $3$ | $6$ | $12$ | $C_6 \times C_3$ | $[\ ]_{3}^{6}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.20.2t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\sqrt{-5}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.40.2t1.b.a | $1$ | $ 2^{3} \cdot 5 $ | \(\Q(\sqrt{-10}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| 1.760.6t1.b.a | $1$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.8340544000.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.152.6t1.b.a | $1$ | $ 2^{3} \cdot 19 $ | 6.6.66724352.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.760.6t1.b.b | $1$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.8340544000.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.380.6t1.b.a | $1$ | $ 2^{2} \cdot 5 \cdot 19 $ | 6.0.1042568000.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
| * | 1.380.6t1.b.b | $1$ | $ 2^{2} \cdot 5 \cdot 19 $ | 6.0.1042568000.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
| 1.152.6t1.b.b | $1$ | $ 2^{3} \cdot 19 $ | 6.6.66724352.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| * | 2.14440.3t2.a.a | $2$ | $ 2^{3} \cdot 5 \cdot 19^{2}$ | 3.1.14440.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.57760.6t3.d.a | $2$ | $ 2^{5} \cdot 5 \cdot 19^{2}$ | 6.2.6672435200.7 | $D_{6}$ (as 6T3) | $1$ | $0$ |
| * | 2.3040.12t18.d.a | $2$ | $ 2^{5} \cdot 5 \cdot 19 $ | 18.0.4641657809149437673472000000000.1 | $S_3 \times C_6$ (as 18T6) | $0$ | $0$ |
| * | 2.3040.12t18.d.b | $2$ | $ 2^{5} \cdot 5 \cdot 19 $ | 18.0.4641657809149437673472000000000.1 | $S_3 \times C_6$ (as 18T6) | $0$ | $0$ |
| * | 2.760.6t5.a.a | $2$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.23104000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.760.6t5.a.b | $2$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.23104000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.