Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 2*x^16 - 5*x^15 + 15*x^14 - 6*x^13 + 2*x^12 - 29*x^11 + 17*x^10 + 11*x^9 + 17*x^8 - 29*x^7 + 2*x^6 - 6*x^5 + 15*x^4 - 5*x^3 + 2*x^2 - 3*x + 1)
gp: K = bnfinit(y^18 - 3*y^17 + 2*y^16 - 5*y^15 + 15*y^14 - 6*y^13 + 2*y^12 - 29*y^11 + 17*y^10 + 11*y^9 + 17*y^8 - 29*y^7 + 2*y^6 - 6*y^5 + 15*y^4 - 5*y^3 + 2*y^2 - 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 + 2*x^16 - 5*x^15 + 15*x^14 - 6*x^13 + 2*x^12 - 29*x^11 + 17*x^10 + 11*x^9 + 17*x^8 - 29*x^7 + 2*x^6 - 6*x^5 + 15*x^4 - 5*x^3 + 2*x^2 - 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 + 2*x^16 - 5*x^15 + 15*x^14 - 6*x^13 + 2*x^12 - 29*x^11 + 17*x^10 + 11*x^9 + 17*x^8 - 29*x^7 + 2*x^6 - 6*x^5 + 15*x^4 - 5*x^3 + 2*x^2 - 3*x + 1)
\( x^{18} - 3 x^{17} + 2 x^{16} - 5 x^{15} + 15 x^{14} - 6 x^{13} + 2 x^{12} - 29 x^{11} + 17 x^{10} + \cdots + 1 \)
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: | $[2, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(4837319251866194741\)
\(\medspace = 23^{6}\cdot 59^{2}\cdot 149\cdot 251^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.92\) | 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: | $23^{1/2}59^{1/2}149^{1/2}251^{1/2}\approx 7123.934516824253$ | ||
| Ramified primes: |
\(23\), \(59\), \(149\), \(251\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{149}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{649}a^{16}-\frac{153}{649}a^{15}+\frac{4}{11}a^{14}-\frac{206}{649}a^{13}+\frac{16}{59}a^{12}-\frac{240}{649}a^{11}+\frac{131}{649}a^{10}+\frac{31}{649}a^{9}-\frac{221}{649}a^{8}+\frac{31}{649}a^{7}+\frac{131}{649}a^{6}-\frac{240}{649}a^{5}+\frac{16}{59}a^{4}-\frac{206}{649}a^{3}+\frac{4}{11}a^{2}-\frac{153}{649}a+\frac{1}{649}$, $\frac{1}{649}a^{17}+\frac{191}{649}a^{15}+\frac{207}{649}a^{14}-\frac{190}{649}a^{13}+\frac{79}{649}a^{12}-\frac{245}{649}a^{11}-\frac{45}{649}a^{10}-\frac{21}{649}a^{9}-\frac{34}{649}a^{8}-\frac{318}{649}a^{7}-\frac{316}{649}a^{6}-\frac{200}{649}a^{5}+\frac{113}{649}a^{4}-\frac{130}{649}a^{3}+\frac{260}{649}a^{2}-\frac{4}{59}a+\frac{153}{649}$
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 $1$
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: | $9$ | 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: |
$\frac{1143}{59}a^{17}-\frac{2806}{59}a^{16}+\frac{755}{59}a^{15}-\frac{5299}{59}a^{14}+\frac{14241}{59}a^{13}+\frac{945}{59}a^{12}+\frac{2766}{59}a^{11}-\frac{31568}{59}a^{10}+\frac{2055}{59}a^{9}+\frac{13743}{59}a^{8}+\frac{26850}{59}a^{7}-\frac{18178}{59}a^{6}-\frac{7690}{59}a^{5}-\frac{11110}{59}a^{4}+\frac{10841}{59}a^{3}+\frac{351}{59}a^{2}+\frac{2488}{59}a-\frac{1977}{59}$, $a$, $\frac{7134}{649}a^{17}-\frac{17297}{649}a^{16}+\frac{4056}{649}a^{15}-\frac{32718}{649}a^{14}+\frac{8008}{59}a^{13}+\frac{8878}{649}a^{12}+\frac{16428}{649}a^{11}-\frac{197968}{649}a^{10}+\frac{7760}{649}a^{9}+\frac{89110}{649}a^{8}+\frac{172144}{649}a^{7}-\frac{112892}{649}a^{6}-\frac{4958}{59}a^{5}-\frac{69172}{649}a^{4}+\frac{69616}{649}a^{3}+\frac{4010}{649}a^{2}+\frac{14966}{649}a-\frac{1170}{59}$, $\frac{13470}{649}a^{17}-\frac{32492}{649}a^{16}+\frac{7858}{649}a^{15}-\frac{62944}{649}a^{14}+\frac{165417}{649}a^{13}+\frac{16391}{649}a^{12}+\frac{37353}{649}a^{11}-\frac{370224}{649}a^{10}+\frac{10474}{649}a^{9}+\frac{154223}{649}a^{8}+\frac{323138}{649}a^{7}-\frac{199931}{649}a^{6}-\frac{93112}{649}a^{5}-\frac{138285}{649}a^{4}+\frac{123427}{649}a^{3}+\frac{5860}{649}a^{2}+\frac{30945}{649}a-\frac{23071}{649}$, $\frac{6705}{649}a^{17}-\frac{16760}{649}a^{16}+\frac{4802}{649}a^{15}-\frac{30484}{649}a^{14}+\frac{84936}{649}a^{13}+\frac{3301}{649}a^{12}+\frac{12123}{649}a^{11}-\frac{187494}{649}a^{10}+\frac{19787}{649}a^{9}+\frac{88859}{649}a^{8}+\frac{155824}{649}a^{7}-\frac{117906}{649}a^{6}-\frac{47645}{649}a^{5}-\frac{60130}{649}a^{4}+\frac{67982}{649}a^{3}+\frac{1679}{649}a^{2}+\frac{13985}{649}a-\frac{12421}{649}$, $\frac{8175}{649}a^{17}-\frac{19983}{649}a^{16}+\frac{5083}{649}a^{15}-\frac{37714}{649}a^{14}+\frac{101591}{649}a^{13}+\frac{8430}{649}a^{12}+\frac{19219}{649}a^{11}-\frac{226749}{649}a^{10}+\frac{11016}{649}a^{9}+\frac{100215}{649}a^{8}+\frac{195915}{649}a^{7}-\frac{127191}{649}a^{6}-\frac{58769}{649}a^{5}-\frac{82252}{649}a^{4}+\frac{76785}{649}a^{3}+\frac{4863}{649}a^{2}+\frac{18627}{649}a-\frac{13990}{649}$, $\frac{4869}{649}a^{17}-\frac{12306}{649}a^{16}+\frac{3925}{649}a^{15}-\frac{22670}{649}a^{14}+\frac{62710}{649}a^{13}+\frac{300}{649}a^{12}+\frac{10831}{649}a^{11}-\frac{137950}{649}a^{10}+\frac{17293}{649}a^{9}+\frac{60622}{649}a^{8}+\frac{115171}{649}a^{7}-\frac{86112}{649}a^{6}-\frac{31611}{649}a^{5}-\frac{47026}{649}a^{4}+\frac{51767}{649}a^{3}+\frac{449}{649}a^{2}+\frac{11685}{649}a-\frac{9805}{649}$, $\frac{2865}{649}a^{17}-\frac{6697}{649}a^{16}+\frac{116}{59}a^{15}-\frac{1208}{59}a^{14}+\frac{33717}{649}a^{13}+\frac{5587}{649}a^{12}+13a^{11}-\frac{75566}{649}a^{10}-\frac{2331}{649}a^{9}+\frac{30760}{649}a^{8}+\frac{66397}{649}a^{7}-\frac{37486}{649}a^{6}-\frac{18398}{649}a^{5}-\frac{29399}{649}a^{4}+\frac{21950}{649}a^{3}+\frac{1618}{649}a^{2}+\frac{6206}{649}a-\frac{4480}{649}$, $\frac{8663}{649}a^{17}-\frac{1916}{59}a^{16}+\frac{5271}{649}a^{15}-\frac{40184}{649}a^{14}+\frac{107474}{649}a^{13}+\frac{9076}{649}a^{12}+\frac{21795}{649}a^{11}-\frac{240675}{649}a^{10}+\frac{11666}{649}a^{9}+\frac{104516}{649}a^{8}+\frac{208687}{649}a^{7}-\frac{136426}{649}a^{6}-\frac{62789}{649}a^{5}-\frac{85782}{649}a^{4}+\frac{84041}{649}a^{3}+\frac{3595}{649}a^{2}+\frac{1728}{59}a-\frac{15703}{649}$
| 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: | \( 67.155663406 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 67.155663406 \cdot 1}{2\cdot\sqrt{4837319251866194741}}\cr\approx \mathstrut & 0.14833690414 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 2*x^16 - 5*x^15 + 15*x^14 - 6*x^13 + 2*x^12 - 29*x^11 + 17*x^10 + 11*x^9 + 17*x^8 - 29*x^7 + 2*x^6 - 6*x^5 + 15*x^4 - 5*x^3 + 2*x^2 - 3*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 - 3*x^17 + 2*x^16 - 5*x^15 + 15*x^14 - 6*x^13 + 2*x^12 - 29*x^11 + 17*x^10 + 11*x^9 + 17*x^8 - 29*x^7 + 2*x^6 - 6*x^5 + 15*x^4 - 5*x^3 + 2*x^2 - 3*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 - 3*x^17 + 2*x^16 - 5*x^15 + 15*x^14 - 6*x^13 + 2*x^12 - 29*x^11 + 17*x^10 + 11*x^9 + 17*x^8 - 29*x^7 + 2*x^6 - 6*x^5 + 15*x^4 - 5*x^3 + 2*x^2 - 3*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 - 3*x^17 + 2*x^16 - 5*x^15 + 15*x^14 - 6*x^13 + 2*x^12 - 29*x^11 + 17*x^10 + 11*x^9 + 17*x^8 - 29*x^7 + 2*x^6 - 6*x^5 + 15*x^4 - 5*x^3 + 2*x^2 - 3*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
$C_2\times S_4^3.S_4$ (as 18T912):
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 663552 |
| The 330 conjugacy class representatives for $C_2\times S_4^3.S_4$ |
| Character table for $C_2\times S_4^3.S_4$ |
Intermediate fields
| 3.1.23.1, 9.3.180181103.1 |
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
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $18$ | $18$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | R |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.8.4.1 | $x^{8} + 98 x^{6} + 38 x^{5} + 3331 x^{4} - 1634 x^{3} + 44919 x^{2} - 57494 x + 224528$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(59\)
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 59.2.0.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 59.2.0.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 59.2.0.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 59.2.0.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 59.2.0.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 59.4.2.1 | $x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(149\)
| 149.2.1.1 | $x^{2} + 149$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 149.2.0.1 | $x^{2} + 145 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 149.2.0.1 | $x^{2} + 145 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 149.6.0.1 | $x^{6} + x^{4} + 105 x^{3} + 33 x^{2} + 55 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 149.6.0.1 | $x^{6} + x^{4} + 105 x^{3} + 33 x^{2} + 55 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(251\)
| $\Q_{251}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{251}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |