Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 150*x^10 + 168*x^9 + 6499*x^8 + 17779*x^7 + 113305*x^6 + 399401*x^5 + 930866*x^4 + 2425574*x^3 + 5043531*x^2 + 4720833*x + 2845051)
gp: K = bnfinit(y^12 - y^11 + 150*y^10 + 168*y^9 + 6499*y^8 + 17779*y^7 + 113305*y^6 + 399401*y^5 + 930866*y^4 + 2425574*y^3 + 5043531*y^2 + 4720833*y + 2845051, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 150*x^10 + 168*x^9 + 6499*x^8 + 17779*x^7 + 113305*x^6 + 399401*x^5 + 930866*x^4 + 2425574*x^3 + 5043531*x^2 + 4720833*x + 2845051);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 + 150*x^10 + 168*x^9 + 6499*x^8 + 17779*x^7 + 113305*x^6 + 399401*x^5 + 930866*x^4 + 2425574*x^3 + 5043531*x^2 + 4720833*x + 2845051)
\( x^{12} - x^{11} + 150 x^{10} + 168 x^{9} + 6499 x^{8} + 17779 x^{7} + 113305 x^{6} + 399401 x^{5} + \cdots + 2845051 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(4294498872017628501555997\)
\(\medspace = 17^{6}\cdot 37^{11}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(112.91\) | 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: | $17^{1/2}37^{11/12}\approx 112.9126779663512$ | ||
| Ramified primes: |
\(17\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{37}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(629=17\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{629}(288,·)$, $\chi_{629}(1,·)$, $\chi_{629}(443,·)$, $\chi_{629}(356,·)$, $\chi_{629}(545,·)$, $\chi_{629}(137,·)$, $\chi_{629}(458,·)$, $\chi_{629}(526,·)$, $\chi_{629}(528,·)$, $\chi_{629}(339,·)$, $\chi_{629}(307,·)$, $\chi_{629}(475,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | 4.0.14638717.1$^{2}$, 12.0.4294498872017628501555997.1$^{30}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{11}a^{7}+\frac{5}{11}a^{6}-\frac{2}{11}a^{5}+\frac{5}{11}a^{4}-\frac{5}{11}a^{3}+\frac{1}{11}a^{2}-\frac{5}{11}a$, $\frac{1}{11}a^{8}-\frac{5}{11}a^{6}+\frac{4}{11}a^{5}+\frac{3}{11}a^{4}+\frac{4}{11}a^{3}+\frac{1}{11}a^{2}+\frac{3}{11}a$, $\frac{1}{33}a^{9}+\frac{1}{33}a^{8}-\frac{1}{33}a^{7}+\frac{8}{33}a^{6}-\frac{4}{11}a^{5}-\frac{2}{11}a^{4}-\frac{5}{11}a^{3}+\frac{8}{33}a^{2}-\frac{2}{11}a+\frac{1}{3}$, $\frac{1}{33}a^{10}+\frac{1}{33}a^{8}-\frac{14}{33}a^{6}+\frac{1}{11}a^{5}-\frac{4}{11}a^{4}+\frac{14}{33}a^{3}+\frac{13}{33}a^{2}+\frac{5}{33}a-\frac{1}{3}$, $\frac{1}{12\!\cdots\!99}a^{11}+\frac{46\!\cdots\!73}{41\!\cdots\!33}a^{10}-\frac{43\!\cdots\!67}{12\!\cdots\!99}a^{9}+\frac{46\!\cdots\!92}{12\!\cdots\!99}a^{8}-\frac{23\!\cdots\!12}{41\!\cdots\!33}a^{7}+\frac{39\!\cdots\!58}{12\!\cdots\!99}a^{6}+\frac{22\!\cdots\!71}{41\!\cdots\!33}a^{5}-\frac{19\!\cdots\!61}{12\!\cdots\!99}a^{4}+\frac{38\!\cdots\!31}{12\!\cdots\!99}a^{3}-\frac{95\!\cdots\!41}{26\!\cdots\!17}a^{2}-\frac{22\!\cdots\!61}{12\!\cdots\!99}a-\frac{16\!\cdots\!04}{23\!\cdots\!47}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $11$ |
Class group and class number
$C_{19546}$, which has order $19546$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $19546$ (assuming GRH)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $5$ | 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: |
$\frac{66\!\cdots\!00}{12\!\cdots\!99}a^{11}-\frac{63\!\cdots\!14}{41\!\cdots\!33}a^{10}+\frac{10\!\cdots\!00}{12\!\cdots\!99}a^{9}-\frac{78\!\cdots\!95}{12\!\cdots\!99}a^{8}+\frac{13\!\cdots\!15}{41\!\cdots\!33}a^{7}+\frac{32\!\cdots\!35}{12\!\cdots\!99}a^{6}+\frac{17\!\cdots\!58}{41\!\cdots\!33}a^{5}+\frac{10\!\cdots\!45}{12\!\cdots\!99}a^{4}+\frac{10\!\cdots\!85}{12\!\cdots\!99}a^{3}+\frac{10\!\cdots\!15}{26\!\cdots\!17}a^{2}+\frac{50\!\cdots\!42}{12\!\cdots\!99}a-\frac{39\!\cdots\!88}{23\!\cdots\!47}$, $\frac{19\!\cdots\!60}{12\!\cdots\!99}a^{11}-\frac{61\!\cdots\!25}{12\!\cdots\!99}a^{10}+\frac{29\!\cdots\!93}{12\!\cdots\!99}a^{9}-\frac{31\!\cdots\!30}{12\!\cdots\!99}a^{8}+\frac{41\!\cdots\!21}{41\!\cdots\!33}a^{7}+\frac{25\!\cdots\!58}{41\!\cdots\!33}a^{6}+\frac{56\!\cdots\!61}{41\!\cdots\!33}a^{5}+\frac{37\!\cdots\!46}{12\!\cdots\!99}a^{4}+\frac{18\!\cdots\!66}{41\!\cdots\!33}a^{3}+\frac{50\!\cdots\!87}{26\!\cdots\!17}a^{2}+\frac{80\!\cdots\!81}{41\!\cdots\!33}a-\frac{34\!\cdots\!64}{79\!\cdots\!49}$, $\frac{11\!\cdots\!49}{11\!\cdots\!09}a^{11}-\frac{12\!\cdots\!26}{33\!\cdots\!73}a^{10}+\frac{17\!\cdots\!33}{11\!\cdots\!09}a^{9}-\frac{81\!\cdots\!36}{37\!\cdots\!03}a^{8}+\frac{70\!\cdots\!24}{11\!\cdots\!09}a^{7}+\frac{64\!\cdots\!65}{37\!\cdots\!03}a^{6}+\frac{27\!\cdots\!88}{37\!\cdots\!03}a^{5}+\frac{15\!\cdots\!67}{11\!\cdots\!09}a^{4}+\frac{34\!\cdots\!23}{11\!\cdots\!09}a^{3}+\frac{12\!\cdots\!68}{23\!\cdots\!47}a^{2}+\frac{60\!\cdots\!17}{11\!\cdots\!09}a-\frac{47\!\cdots\!47}{72\!\cdots\!59}$, $\frac{66\!\cdots\!21}{12\!\cdots\!99}a^{11}-\frac{16\!\cdots\!87}{12\!\cdots\!99}a^{10}+\frac{34\!\cdots\!10}{41\!\cdots\!33}a^{9}-\frac{46\!\cdots\!42}{12\!\cdots\!99}a^{8}+\frac{47\!\cdots\!99}{12\!\cdots\!99}a^{7}+\frac{18\!\cdots\!43}{41\!\cdots\!33}a^{6}+\frac{26\!\cdots\!93}{41\!\cdots\!33}a^{5}+\frac{19\!\cdots\!09}{12\!\cdots\!99}a^{4}+\frac{50\!\cdots\!45}{12\!\cdots\!99}a^{3}+\frac{36\!\cdots\!39}{26\!\cdots\!17}a^{2}+\frac{17\!\cdots\!56}{12\!\cdots\!99}a+\frac{33\!\cdots\!51}{79\!\cdots\!49}$, $\frac{32\!\cdots\!98}{37\!\cdots\!03}a^{11}-\frac{25\!\cdots\!72}{10\!\cdots\!19}a^{10}+\frac{14\!\cdots\!94}{11\!\cdots\!09}a^{9}-\frac{42\!\cdots\!20}{37\!\cdots\!03}a^{8}+\frac{62\!\cdots\!46}{11\!\cdots\!09}a^{7}+\frac{49\!\cdots\!10}{11\!\cdots\!09}a^{6}+\frac{30\!\cdots\!56}{37\!\cdots\!03}a^{5}+\frac{67\!\cdots\!93}{37\!\cdots\!03}a^{4}+\frac{36\!\cdots\!03}{11\!\cdots\!09}a^{3}+\frac{30\!\cdots\!32}{23\!\cdots\!47}a^{2}+\frac{14\!\cdots\!14}{11\!\cdots\!09}a+\frac{89\!\cdots\!88}{21\!\cdots\!77}$
| 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: | \( 2518.23324049 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 2518.23324049 \cdot 19546}{2\cdot\sqrt{4294498872017628501555997}}\cr\approx \mathstrut & 0.730712967737 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 150*x^10 + 168*x^9 + 6499*x^8 + 17779*x^7 + 113305*x^6 + 399401*x^5 + 930866*x^4 + 2425574*x^3 + 5043531*x^2 + 4720833*x + 2845051)
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^12 - x^11 + 150*x^10 + 168*x^9 + 6499*x^8 + 17779*x^7 + 113305*x^6 + 399401*x^5 + 930866*x^4 + 2425574*x^3 + 5043531*x^2 + 4720833*x + 2845051, 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^12 - x^11 + 150*x^10 + 168*x^9 + 6499*x^8 + 17779*x^7 + 113305*x^6 + 399401*x^5 + 930866*x^4 + 2425574*x^3 + 5043531*x^2 + 4720833*x + 2845051);
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^12 - x^11 + 150*x^10 + 168*x^9 + 6499*x^8 + 17779*x^7 + 113305*x^6 + 399401*x^5 + 930866*x^4 + 2425574*x^3 + 5043531*x^2 + 4720833*x + 2845051);
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 cyclic group of order 12 |
| The 12 conjugacy class representatives for $C_{12}$ |
| Character table for $C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{37}) \), 3.3.1369.1, 4.0.14638717.1, 6.6.69343957.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.12.0.1}{12} }$ | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.1.0.1}{1} }^{12}$ | ${\href{/padicField/13.12.0.1}{12} }$ | R | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.1.0.1}{1} }^{12}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.12.6.2 | $x^{12} + 578 x^{8} + 835210 x^{4} - 4259571 x^{2} + 72412707$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |
|
\(37\)
| 37.12.11.1 | $x^{12} + 407$ | $12$ | $1$ | $11$ | $C_{12}$ | $[\ ]_{12}$ |