Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 - 4*x^14 - 11*x^12 + 4*x^10 - 4*x^8 + 33*x^6 - 32*x^4 + 10*x^2 - 1)
gp: K = bnfinit(y^18 + 3*y^16 - 4*y^14 - 11*y^12 + 4*y^10 - 4*y^8 + 33*y^6 - 32*y^4 + 10*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 3*x^16 - 4*x^14 - 11*x^12 + 4*x^10 - 4*x^8 + 33*x^6 - 32*x^4 + 10*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 3*x^16 - 4*x^14 - 11*x^12 + 4*x^10 - 4*x^8 + 33*x^6 - 32*x^4 + 10*x^2 - 1)
\( x^{18} + 3x^{16} - 4x^{14} - 11x^{12} + 4x^{10} - 4x^{8} + 33x^{6} - 32x^{4} + 10x^{2} - 1 \)
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: | $[10, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(53977668015744910360576\)
\(\medspace = 2^{18}\cdot 453771377^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.32\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(453771377\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{101}a^{16}+\frac{11}{101}a^{14}-\frac{17}{101}a^{12}-\frac{46}{101}a^{10}+\frac{40}{101}a^{8}+\frac{13}{101}a^{6}+\frac{36}{101}a^{4}-\frac{47}{101}a^{2}+\frac{38}{101}$, $\frac{1}{101}a^{17}+\frac{11}{101}a^{15}-\frac{17}{101}a^{13}-\frac{46}{101}a^{11}+\frac{40}{101}a^{9}+\frac{13}{101}a^{7}+\frac{36}{101}a^{5}-\frac{47}{101}a^{3}+\frac{38}{101}a$
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: | $13$ | 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: |
$\frac{651}{101}a^{16}+\frac{2212}{101}a^{14}-\frac{1674}{101}a^{12}-\frac{7625}{101}a^{10}-\frac{422}{101}a^{8}-\frac{3354}{101}a^{6}+\frac{19699}{101}a^{4}-\frac{13528}{101}a^{2}+\frac{2316}{101}$, $\frac{629}{101}a^{16}+\frac{1869}{101}a^{14}-\frac{2714}{101}a^{12}-\frac{7421}{101}a^{10}+\frac{2738}{101}a^{8}-\frac{913}{101}a^{6}+\frac{21937}{101}a^{4}-\frac{19261}{101}a^{2}+\frac{3399}{101}$, $a$, $\frac{81}{101}a^{16}+\frac{83}{101}a^{14}-\frac{973}{101}a^{12}-\frac{898}{101}a^{10}+\frac{2230}{101}a^{8}+\frac{1053}{101}a^{6}+\frac{4330}{101}a^{4}-\frac{6433}{101}a^{2}+\frac{1462}{101}$, $\frac{730}{101}a^{17}+\frac{2172}{101}a^{15}-\frac{3118}{101}a^{13}-\frac{8532}{101}a^{11}+\frac{3142}{101}a^{9}-\frac{1317}{101}a^{7}+\frac{25270}{101}a^{5}-\frac{22493}{101}a^{3}+\frac{4308}{101}a$, $\frac{1361}{101}a^{17}+\frac{4164}{101}a^{15}-\frac{5361}{101}a^{13}-\frac{15944}{101}a^{11}+\frac{4546}{101}a^{9}-\frac{3214}{101}a^{7}+\frac{45966}{101}a^{5}-\frac{39222}{101}a^{3}+\frac{7177}{101}a$, $\frac{1343}{101}a^{16}+\frac{4572}{101}a^{14}-\frac{3439}{101}a^{12}-\frac{15823}{101}a^{10}-\frac{1022}{101}a^{8}-\frac{6781}{101}a^{6}+\frac{40874}{101}a^{4}-\frac{27266}{101}a^{2}-a+\frac{4372}{101}$, $\frac{1361}{101}a^{17}+\frac{710}{101}a^{16}+\frac{4164}{101}a^{15}+\frac{1952}{101}a^{14}-\frac{5361}{101}a^{13}-\frac{3687}{101}a^{12}-\frac{15944}{101}a^{11}-\frac{8319}{101}a^{10}+\frac{4546}{101}a^{9}+\frac{4968}{101}a^{8}-\frac{3214}{101}a^{7}+\frac{140}{101}a^{6}+\frac{45966}{101}a^{5}+\frac{26267}{101}a^{4}-\frac{39222}{101}a^{3}-\frac{25694}{101}a^{2}+\frac{7177}{101}a+\frac{4760}{101}$, $\frac{730}{101}a^{17}-\frac{631}{101}a^{16}+\frac{2172}{101}a^{15}-\frac{1992}{101}a^{14}-\frac{3118}{101}a^{13}+\frac{2243}{101}a^{12}-\frac{8532}{101}a^{11}+\frac{7412}{101}a^{10}+\frac{3142}{101}a^{9}-\frac{1404}{101}a^{8}-\frac{1317}{101}a^{7}+\frac{1897}{101}a^{6}+\frac{25270}{101}a^{5}-\frac{20696}{101}a^{4}-\frac{22493}{101}a^{3}+\frac{16729}{101}a^{2}+\frac{4308}{101}a-\frac{2869}{101}$, $\frac{2276}{101}a^{17}-\frac{144}{101}a^{16}+\frac{7664}{101}a^{15}-\frac{574}{101}a^{14}-\frac{6170}{101}a^{13}+\frac{24}{101}a^{12}-\frac{26825}{101}a^{11}+\frac{1675}{101}a^{10}-\frac{769}{101}a^{9}+\frac{1108}{101}a^{8}-\frac{10812}{101}a^{7}+\frac{1461}{101}a^{6}+\frac{70220}{101}a^{5}-\frac{3568}{101}a^{4}-\frac{48190}{101}a^{3}+\frac{1011}{101}a^{2}+\frac{7809}{101}a-\frac{18}{101}$, $\frac{81}{101}a^{17}+\frac{694}{101}a^{16}+\frac{83}{101}a^{15}+\frac{2483}{101}a^{14}-\frac{973}{101}a^{13}-\frac{1294}{101}a^{12}-\frac{898}{101}a^{11}-\frac{8189}{101}a^{10}+\frac{2230}{101}a^{9}-\frac{1934}{101}a^{8}+\frac{1053}{101}a^{7}-\frac{4411}{101}a^{6}+\frac{4330}{101}a^{5}+\frac{19934}{101}a^{4}-\frac{6433}{101}a^{3}-\frac{11206}{101}a^{2}+\frac{1462}{101}a+\frac{1526}{101}$, $\frac{692}{101}a^{17}+\frac{2360}{101}a^{15}-\frac{1765}{101}a^{13}-\frac{8198}{101}a^{11}-\frac{600}{101}a^{9}-\frac{3427}{101}a^{7}+\frac{21175}{101}a^{5}-\frac{13738}{101}a^{3}+\frac{2056}{101}a+1$, $\frac{611}{101}a^{17}+\frac{20}{101}a^{16}+\frac{1671}{101}a^{15}+\frac{220}{101}a^{14}-\frac{3216}{101}a^{13}+\frac{569}{101}a^{12}-\frac{7199}{101}a^{11}-\frac{213}{101}a^{10}+\frac{4341}{101}a^{9}-\frac{1826}{101}a^{8}+\frac{267}{101}a^{7}-\frac{1457}{101}a^{6}+\frac{22804}{101}a^{5}-\frac{997}{101}a^{4}-\frac{22253}{101}a^{3}+\frac{3201}{101}a^{2}+\frac{4129}{101}a-\frac{553}{101}$
| 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: | \( 45411.702991 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{4}\cdot 45411.702991 \cdot 1}{2\cdot\sqrt{53977668015744910360576}}\cr\approx \mathstrut & 0.15597326530 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 - 4*x^14 - 11*x^12 + 4*x^10 - 4*x^8 + 33*x^6 - 32*x^4 + 10*x^2 - 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^16 - 4*x^14 - 11*x^12 + 4*x^10 - 4*x^8 + 33*x^6 - 32*x^4 + 10*x^2 - 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^16 - 4*x^14 - 11*x^12 + 4*x^10 - 4*x^8 + 33*x^6 - 32*x^4 + 10*x^2 - 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^16 - 4*x^14 - 11*x^12 + 4*x^10 - 4*x^8 + 33*x^6 - 32*x^4 + 10*x^2 - 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
$C_2^8.S_9$ (as 18T964):
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 non-solvable group of order 92897280 |
| The 150 conjugacy class representatives for $C_2^8.S_9$ |
| Character table for $C_2^8.S_9$ |
Intermediate fields
| 9.5.453771377.1 |
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 sibling: | data not computed |
| Degree 36 sibling: | 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 | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.7.0.1}{7} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.9.0.1}{9} }^{2}$ | ${\href{/padicField/19.9.0.1}{9} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.18.49 | $x^{18} + 18 x^{17} + 198 x^{16} + 1656 x^{15} + 10952 x^{14} + 59200 x^{13} + 271888 x^{12} + 1046816 x^{11} + 3433408 x^{10} + 9627328 x^{9} + 22512896 x^{8} + 44617856 x^{7} + 74973568 x^{6} + 95518720 x^{5} + 114645760 x^{4} + 72416768 x^{3} + 60099328 x^{2} + 7582208 x + 5847552$ | $2$ | $9$ | $18$ | 18T177 | $[2, 2, 2, 2, 2, 2]^{9}$ |
|
\(453771377\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |