Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 5*x^16 - 4*x^15 + 4*x^14 - 4*x^11 + x^10 - 6*x^9 - 5*x^8 - 2*x^7 + 16*x^4 + 24*x^3 + 17*x^2 + 6*x + 1)
gp: K = bnfinit(y^18 - 2*y^17 + 5*y^16 - 4*y^15 + 4*y^14 - 4*y^11 + y^10 - 6*y^9 - 5*y^8 - 2*y^7 + 16*y^4 + 24*y^3 + 17*y^2 + 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 + 5*x^16 - 4*x^15 + 4*x^14 - 4*x^11 + x^10 - 6*x^9 - 5*x^8 - 2*x^7 + 16*x^4 + 24*x^3 + 17*x^2 + 6*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 2*x^17 + 5*x^16 - 4*x^15 + 4*x^14 - 4*x^11 + x^10 - 6*x^9 - 5*x^8 - 2*x^7 + 16*x^4 + 24*x^3 + 17*x^2 + 6*x + 1)
\( x^{18} - 2 x^{17} + 5 x^{16} - 4 x^{15} + 4 x^{14} - 4 x^{11} + x^{10} - 6 x^{9} - 5 x^{8} - 2 x^{7} + \cdots + 1 \)
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: |
\(-1224371671109336996968071168\)
\(\medspace = -\,2^{16}\cdot 3^{9}\cdot 7^{16}\cdot 13^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(31.98\) | 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: |
\(2\), \(3\), \(7\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7}a^{16}-\frac{1}{7}a^{15}+\frac{2}{7}a^{14}+\frac{3}{7}a^{9}-\frac{3}{7}a^{8}-\frac{1}{7}a^{7}+\frac{2}{7}a^{2}-\frac{2}{7}a-\frac{3}{7}$, $\frac{1}{14}a^{17}-\frac{1}{14}a^{16}+\frac{1}{7}a^{15}-\frac{2}{7}a^{10}-\frac{3}{14}a^{9}-\frac{1}{14}a^{8}+\frac{1}{7}a^{3}-\frac{1}{7}a^{2}-\frac{3}{14}a-\frac{1}{2}$
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
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: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{457}{14} a^{17} - \frac{1185}{14} a^{16} + \frac{1458}{7} a^{15} - 240 a^{14} + 238 a^{13} - 98 a^{12} + 14 a^{11} - \frac{816}{7} a^{10} + \frac{1345}{14} a^{9} - \frac{3271}{14} a^{8} - 43 a^{7} + \frac{3656}{7} a^{3} + \frac{3316}{7} a^{2} + \frac{2689}{14} a + \frac{43}{2} \)
(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: |
$\frac{271}{14}a^{17}-\frac{631}{14}a^{16}+\frac{766}{7}a^{15}-\frac{752}{7}a^{14}+98a^{13}-14a^{12}-14a^{11}-\frac{444}{7}a^{10}+\frac{529}{14}a^{9}-\frac{1683}{14}a^{8}-\frac{457}{7}a^{7}+\frac{2168}{7}a^{3}+\frac{2540}{7}a^{2}+\frac{2455}{14}a+\frac{457}{14}$, $\frac{94}{7}a^{17}-\frac{433}{7}a^{16}+\frac{1122}{7}a^{15}-\frac{1987}{7}a^{14}+328a^{13}-277a^{12}+128a^{11}-\frac{572}{7}a^{10}+\frac{1004}{7}a^{9}-\frac{1359}{7}a^{8}+\frac{1361}{7}a^{7}-2a^{6}+3a^{5}-3a^{4}+\frac{1511}{7}a^{3}-\frac{1671}{7}a^{2}-\frac{1683}{7}a-\frac{691}{7}$, $\frac{41}{2}a^{17}-\frac{1031}{14}a^{16}+\frac{1310}{7}a^{15}-\frac{2032}{7}a^{14}+322a^{13}-238a^{12}+98a^{11}-96a^{10}+\frac{1919}{14}a^{9}-\frac{3067}{14}a^{8}+\frac{918}{7}a^{7}+2a^{6}+327a^{3}-\frac{205}{7}a^{2}-\frac{1767}{14}a-\frac{953}{14}$, $\frac{85}{14}a^{17}-\frac{263}{14}a^{16}+\frac{342}{7}a^{15}-\frac{500}{7}a^{14}+86a^{13}-70a^{12}+47a^{11}-\frac{352}{7}a^{10}+\frac{583}{14}a^{9}-\frac{839}{14}a^{8}+\frac{110}{7}a^{7}-4a^{6}-5a^{5}+11a^{4}+\frac{582}{7}a^{3}+\frac{374}{7}a^{2}+\frac{227}{14}a-\frac{5}{14}$, $\frac{1185}{14}a^{17}-\frac{2917}{14}a^{16}+509a^{15}-\frac{3811}{7}a^{14}+518a^{13}-150a^{12}-20a^{11}-\frac{2006}{7}a^{10}+\frac{2911}{14}a^{9}-\frac{1129}{2}a^{8}-\frac{1423}{7}a^{7}+4a^{6}+3a^{5}-4a^{4}+\frac{9459}{7}a^{3}+\frac{9879}{7}a^{2}+\frac{1257}{2}a+\frac{1353}{14}$, $\frac{5367}{14}a^{17}-\frac{14849}{14}a^{16}+2627a^{15}-\frac{22852}{7}a^{14}+3336a^{13}-1694a^{12}+419a^{11}-\frac{9915}{7}a^{10}+\frac{19111}{14}a^{9}-\frac{5937}{2}a^{8}-\frac{5}{7}a^{7}+24a^{6}-4a^{5}-4a^{4}+\frac{42957}{7}a^{3}+\frac{31470}{7}a^{2}+\frac{2841}{2}a-\frac{401}{14}$, $\frac{8}{7}a^{17}-\frac{208}{7}a^{16}+\frac{566}{7}a^{15}-\frac{1331}{7}a^{14}+232a^{13}-230a^{12}+111a^{11}-\frac{193}{7}a^{10}+\frac{685}{7}a^{9}-\frac{696}{7}a^{8}+\frac{1411}{7}a^{7}+7a^{6}-2a^{5}+3a^{4}+\frac{142}{7}a^{3}-\frac{2894}{7}a^{2}-\frac{2221}{7}a-\frac{737}{7}$, $\frac{111}{14}a^{17}-\frac{375}{14}a^{16}+\frac{474}{7}a^{15}-\frac{705}{7}a^{14}+110a^{13}-77a^{12}+29a^{11}-\frac{243}{7}a^{10}+\frac{625}{14}a^{9}-\frac{1139}{14}a^{8}+\frac{251}{7}a^{7}-4a^{6}-5a^{5}-4a^{4}+\frac{846}{7}a^{3}+\frac{80}{7}a^{2}-\frac{421}{14}a-\frac{279}{14}$
| 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: | \( 8118260.80498 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 8118260.80498 \cdot 1}{6\cdot\sqrt{1224371671109336996968071168}}\cr\approx \mathstrut & 0.590165649409 \end{aligned}\] (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 + 5*x^16 - 4*x^15 + 4*x^14 - 4*x^11 + x^10 - 6*x^9 - 5*x^8 - 2*x^7 + 16*x^4 + 24*x^3 + 17*x^2 + 6*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 - 2*x^17 + 5*x^16 - 4*x^15 + 4*x^14 - 4*x^11 + x^10 - 6*x^9 - 5*x^8 - 2*x^7 + 16*x^4 + 24*x^3 + 17*x^2 + 6*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 - 2*x^17 + 5*x^16 - 4*x^15 + 4*x^14 - 4*x^11 + x^10 - 6*x^9 - 5*x^8 - 2*x^7 + 16*x^4 + 24*x^3 + 17*x^2 + 6*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 - 2*x^17 + 5*x^16 - 4*x^15 + 4*x^14 - 4*x^11 + x^10 - 6*x^9 - 5*x^8 - 2*x^7 + 16*x^4 + 24*x^3 + 17*x^2 + 6*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
$\SL(2,8)^2:C_6$ (as 18T937):
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 1524096 |
| The 42 conjugacy class representatives for $\SL(2,8)^2:C_6$ |
| Character table for $\SL(2,8)^2:C_6$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \) |
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 | R | R | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | $18$ | R | $18$ | ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $18$ | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| Deg $16$ | $8$ | $2$ | $16$ | ||||
|
\(3\)
| Deg $18$ | $2$ | $9$ | $9$ | |||
|
\(7\)
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.7.7.5 | $x^{7} + 7 x + 7$ | $7$ | $1$ | $7$ | $F_7$ | $[7/6]_{6}$ | |
| 7.7.7.1 | $x^{7} + 42 x + 7$ | $7$ | $1$ | $7$ | $F_7$ | $[7/6]_{6}$ | |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |