Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 35*x^15 + 599*x^12 - 6435*x^9 + 33352*x^6 + 7360*x^3 + 512)
gp: K = bnfinit(y^18 - 35*y^15 + 599*y^12 - 6435*y^9 + 33352*y^6 + 7360*y^3 + 512, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 35*x^15 + 599*x^12 - 6435*x^9 + 33352*x^6 + 7360*x^3 + 512);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 35*x^15 + 599*x^12 - 6435*x^9 + 33352*x^6 + 7360*x^3 + 512)
\( x^{18} - 35x^{15} + 599x^{12} - 6435x^{9} + 33352x^{6} + 7360x^{3} + 512 \)
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: |
\(-140704327411684407000000000000\)
\(\medspace = -\,2^{12}\cdot 3^{20}\cdot 5^{12}\cdot 7^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(41.62\) | 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: | $2^{2/3}3^{7/6}5^{2/3}7^{1/2}\approx 44.2442818763163$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{4}a^{11}+\frac{1}{4}a^{8}-\frac{1}{4}a^{5}+\frac{1}{4}a^{2}$, $\frac{1}{72}a^{12}-\frac{3}{8}a^{9}-\frac{1}{8}a^{6}-\frac{3}{8}a^{3}+\frac{2}{9}$, $\frac{1}{144}a^{13}-\frac{3}{16}a^{10}+\frac{7}{16}a^{7}-\frac{3}{16}a^{4}+\frac{1}{9}a$, $\frac{1}{288}a^{14}-\frac{3}{32}a^{11}+\frac{7}{32}a^{8}+\frac{13}{32}a^{5}+\frac{1}{18}a^{2}$, $\frac{1}{3772510272}a^{15}-\frac{11787011}{3772510272}a^{12}+\frac{144812351}{419167808}a^{9}-\frac{187267371}{419167808}a^{6}+\frac{143282045}{471563784}a^{3}-\frac{23585114}{58945473}$, $\frac{1}{22635061632}a^{16}+\frac{1}{11317530816}a^{15}+\frac{1}{864}a^{14}-\frac{11787011}{22635061632}a^{13}+\frac{40608965}{11317530816}a^{12}+\frac{5}{96}a^{11}+\frac{563980159}{2515006848}a^{10}-\frac{143847795}{419167808}a^{9}+\frac{47}{96}a^{8}-\frac{606435179}{2515006848}a^{7}+\frac{199557423}{419167808}a^{6}+\frac{5}{96}a^{5}-\frac{799845523}{2829382704}a^{4}-\frac{252559079}{707345676}a^{3}+\frac{11}{108}a^{2}+\frac{153251305}{353672838}a+\frac{48459353}{176836419}$, $\frac{1}{45270123264}a^{17}+\frac{1}{11317530816}a^{15}-\frac{7131443}{5030013696}a^{14}+\frac{1}{432}a^{13}-\frac{11787011}{11317530816}a^{12}+\frac{302000279}{5030013696}a^{11}+\frac{5}{48}a^{10}+\frac{563980159}{1257503424}a^{9}+\frac{1960967645}{5030013696}a^{8}-\frac{1}{48}a^{7}-\frac{606435179}{1257503424}a^{6}-\frac{136821611}{707345676}a^{5}+\frac{5}{48}a^{4}+\frac{614845829}{1414691352}a^{3}+\frac{4511491}{39296982}a^{2}+\frac{11}{54}a-\frac{23585114}{176836419}$
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
$C_{3}\times C_{6}$, which has order $18$ (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: |
$\frac{2164}{6549497}a^{16}-\frac{151475}{13098994}a^{13}+\frac{1301445}{6549497}a^{10}-\frac{14022200}{6549497}a^{7}+\frac{73259200}{6549497}a^{4}+\frac{19227011}{13098994}a$, $\frac{32940209}{22635061632}a^{17}+\frac{5197963}{11317530816}a^{16}-\frac{1955}{209583904}a^{15}-\frac{1158530459}{22635061632}a^{14}-\frac{60616183}{3772510272}a^{13}+\frac{1358623}{5658765408}a^{12}+\frac{738135085}{838335616}a^{11}+\frac{346462817}{1257503424}a^{10}-\frac{1998835}{628751712}a^{9}-\frac{7976865201}{838335616}a^{8}-\frac{3732675925}{1257503424}a^{7}+\frac{14336015}{628751712}a^{6}+\frac{35486270029}{707345676}a^{5}+\frac{43824325921}{2829382704}a^{4}+\frac{784805}{157187928}a^{3}+\frac{1484540687}{707345676}a^{2}+\frac{239625877}{117890946}a-\frac{244823917}{176836419}$, $\frac{190145}{160532352}a^{16}-\frac{2225873}{53510784}a^{13}+\frac{12737551}{17836928}a^{10}-\frac{137255099}{17836928}a^{7}+\frac{807777895}{20066544}a^{4}+\frac{2208308}{418053}a-1$, $\frac{29200817}{45270123264}a^{17}-\frac{37206371}{22635061632}a^{16}+\frac{1955}{209583904}a^{15}-\frac{1027656059}{45270123264}a^{14}+\frac{435080459}{7545020544}a^{13}-\frac{1358623}{5658765408}a^{12}+\frac{654842605}{1676671232}a^{11}-\frac{2488920325}{2515006848}a^{10}+\frac{1998835}{628751712}a^{9}-\frac{7079444401}{1676671232}a^{8}+\frac{26818320809}{2515006848}a^{7}-\frac{14336015}{628751712}a^{6}+\frac{31530273229}{1414691352}a^{5}-\frac{39430252279}{707345676}a^{4}-\frac{784805}{157187928}a^{3}+\frac{329771057}{353672838}a^{2}-\frac{862368733}{117890946}a+\frac{244823917}{176836419}$, $\frac{6338677}{7545020544}a^{16}-\frac{223131151}{7545020544}a^{13}+\frac{426532507}{838335616}a^{10}-\frac{4610108471}{838335616}a^{7}+\frac{27427546835}{943127568}a^{4}+\frac{25546493}{117890946}a$, $\frac{14563}{2986944}a^{17}-\frac{6338797}{2515006848}a^{16}+\frac{8656}{6549497}a^{15}-\frac{512287}{2986944}a^{14}+\frac{222881119}{2515006848}a^{13}-\frac{302950}{6549497}a^{12}+\frac{2937741}{995648}a^{11}-\frac{1278047761}{838335616}a^{10}+\frac{5205780}{6549497}a^{9}-\frac{31754369}{995648}a^{8}+\frac{13812957861}{838335616}a^{7}-\frac{56088800}{6549497}a^{6}+\frac{251131661}{1493472}a^{5}-\frac{13635887341}{157187928}a^{4}+\frac{293036800}{6549497}a^{3}+\frac{1313237}{186684}a^{2}-\frac{142606463}{39296982}a+\frac{18805531}{6549497}$, $\frac{521835299}{45270123264}a^{17}+\frac{60979555}{22635061632}a^{16}-\frac{509333}{471563784}a^{15}-\frac{18325869473}{45270123264}a^{14}-\frac{79441723}{838335616}a^{13}+\frac{53022701}{1414691352}a^{12}+\frac{11656259927}{1676671232}a^{11}+\frac{4107442829}{2515006848}a^{10}-\frac{100437857}{157187928}a^{9}-\frac{125732989795}{1676671232}a^{8}-\frac{44466333841}{2515006848}a^{7}+\frac{1075764685}{157187928}a^{6}+\frac{556225310101}{1414691352}a^{5}+\frac{264985458437}{2829382704}a^{4}-\frac{2049184777}{58945473}a^{3}+\frac{27408322417}{707345676}a^{2}-\frac{32689312}{19648491}a-\frac{1764053717}{176836419}$, $\frac{32220515}{2515006848}a^{17}+\frac{17715815}{3772510272}a^{16}-\frac{376914059}{838335616}a^{14}-\frac{207498391}{1257503424}a^{13}+\frac{6467251215}{838335616}a^{11}+\frac{1189008649}{419167808}a^{10}-\frac{69685912827}{838335616}a^{8}-\frac{12832919933}{419167808}a^{7}+\frac{34155589879}{78593964}a^{5}+\frac{75948087985}{471563784}a^{4}+\frac{747006667}{13098994}a^{2}+\frac{203085272}{19648491}a-7$
| 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: | \( 5863974.111036323 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 5863974.111036323 \cdot 18}{2\cdot\sqrt{140704327411684407000000000000}}\cr\approx \mathstrut & 2.14733420305874 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 35*x^15 + 599*x^12 - 6435*x^9 + 33352*x^6 + 7360*x^3 + 512)
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 - 35*x^15 + 599*x^12 - 6435*x^9 + 33352*x^6 + 7360*x^3 + 512, 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 - 35*x^15 + 599*x^12 - 6435*x^9 + 33352*x^6 + 7360*x^3 + 512);
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 - 35*x^15 + 599*x^12 - 6435*x^9 + 33352*x^6 + 7360*x^3 + 512);
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 solvable group of order 36 |
| The 12 conjugacy class representatives for $C_6:S_3$ |
| Character table for $C_6:S_3$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), 3.1.108.1, 3.1.300.1, 3.1.675.1, 3.1.2700.1, 6.0.4000752.4, 6.0.156279375.1, 6.0.30870000.1, 6.0.2500470000.2, 9.1.59049000000.1 |
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 18 sibling: | 18.2.422112982235053221000000000000.1 |
| 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 | R | R | R | R | ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(3\)
| 3.6.6.3 | $x^{6} + 18 x^{5} + 120 x^{4} + 386 x^{3} + 723 x^{2} + 732 x + 305$ | $3$ | $2$ | $6$ | $D_{6}$ | $[3/2]_{2}^{2}$ |
| 3.12.14.6 | $x^{12} + 6 x^{8} + 15 x^{6} + 9 x^{4} + 18 x^{2} + 9$ | $6$ | $2$ | $14$ | $D_6$ | $[3/2]_{2}^{2}$ | |
|
\(5\)
| 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(7\)
| 7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |