Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 176*x^10 - 241*x^9 + 13764*x^8 - 20921*x^7 + 610782*x^6 - 868596*x^5 + 16022269*x^4 - 17677113*x^3 + 231278288*x^2 - 141901568*x + 1423267621)
gp: K = bnfinit(y^12 - y^11 + 176*y^10 - 241*y^9 + 13764*y^8 - 20921*y^7 + 610782*y^6 - 868596*y^5 + 16022269*y^4 - 17677113*y^3 + 231278288*y^2 - 141901568*y + 1423267621, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 176*x^10 - 241*x^9 + 13764*x^8 - 20921*x^7 + 610782*x^6 - 868596*x^5 + 16022269*x^4 - 17677113*x^3 + 231278288*x^2 - 141901568*x + 1423267621);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 + 176*x^10 - 241*x^9 + 13764*x^8 - 20921*x^7 + 610782*x^6 - 868596*x^5 + 16022269*x^4 - 17677113*x^3 + 231278288*x^2 - 141901568*x + 1423267621)
\( x^{12} - x^{11} + 176 x^{10} - 241 x^{9} + 13764 x^{8} - 20921 x^{7} + 610782 x^{6} - 868596 x^{5} + \cdots + 1423267621 \)
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: |
\(7787035719905674198640625\)
\(\medspace = 3^{6}\cdot 5^{6}\cdot 7^{8}\cdot 17^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(118.65\) | 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: | $3^{1/2}5^{1/2}7^{2/3}17^{3/4}\approx 118.65362581134335$ | ||
| Ramified primes: |
\(3\), \(5\), \(7\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1785=3\cdot 5\cdot 7\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1785}(256,·)$, $\chi_{1785}(1,·)$, $\chi_{1785}(1619,·)$, $\chi_{1785}(781,·)$, $\chi_{1785}(526,·)$, $\chi_{1785}(16,·)$, $\chi_{1785}(914,·)$, $\chi_{1785}(659,·)$, $\chi_{1785}(149,·)$, $\chi_{1785}(599,·)$, $\chi_{1785}(344,·)$, $\chi_{1785}(1276,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | 4.0.1105425.1$^{2}$, 12.0.7787035719905674198640625.1$^{30}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{94}a^{9}+\frac{35}{94}a^{8}+\frac{35}{94}a^{7}-\frac{23}{94}a^{6}-\frac{14}{47}a^{5}+\frac{23}{94}a^{4}-\frac{15}{94}a^{3}+\frac{27}{94}a^{2}-\frac{43}{94}a+\frac{25}{94}$, $\frac{1}{94}a^{10}+\frac{16}{47}a^{8}-\frac{13}{47}a^{7}+\frac{25}{94}a^{6}-\frac{31}{94}a^{5}+\frac{13}{47}a^{4}-\frac{6}{47}a^{3}+\frac{23}{47}a^{2}+\frac{13}{47}a-\frac{29}{94}$, $\frac{1}{21\!\cdots\!94}a^{11}-\frac{12\!\cdots\!36}{10\!\cdots\!47}a^{10}+\frac{31\!\cdots\!82}{10\!\cdots\!47}a^{9}-\frac{50\!\cdots\!91}{10\!\cdots\!47}a^{8}+\frac{64\!\cdots\!79}{21\!\cdots\!94}a^{7}-\frac{45\!\cdots\!41}{21\!\cdots\!94}a^{6}-\frac{25\!\cdots\!65}{10\!\cdots\!47}a^{5}+\frac{31\!\cdots\!52}{10\!\cdots\!47}a^{4}-\frac{23\!\cdots\!89}{10\!\cdots\!47}a^{3}+\frac{33\!\cdots\!97}{10\!\cdots\!47}a^{2}+\frac{61\!\cdots\!49}{21\!\cdots\!94}a+\frac{11\!\cdots\!95}{24\!\cdots\!29}$
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
$C_{2}\times C_{7466}$, which has order $14932$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $14932$ (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: |
$\frac{50\!\cdots\!95}{22\!\cdots\!01}a^{11}+\frac{51\!\cdots\!59}{22\!\cdots\!01}a^{10}+\frac{10\!\cdots\!48}{22\!\cdots\!01}a^{9}+\frac{65\!\cdots\!36}{22\!\cdots\!01}a^{8}+\frac{78\!\cdots\!31}{22\!\cdots\!01}a^{7}+\frac{34\!\cdots\!58}{22\!\cdots\!01}a^{6}+\frac{29\!\cdots\!13}{22\!\cdots\!01}a^{5}+\frac{99\!\cdots\!01}{22\!\cdots\!01}a^{4}+\frac{57\!\cdots\!24}{22\!\cdots\!01}a^{3}+\frac{16\!\cdots\!72}{22\!\cdots\!01}a^{2}+\frac{48\!\cdots\!78}{22\!\cdots\!01}a+\frac{25\!\cdots\!52}{52\!\cdots\!07}$, $\frac{50\!\cdots\!95}{22\!\cdots\!01}a^{11}+\frac{51\!\cdots\!59}{22\!\cdots\!01}a^{10}+\frac{10\!\cdots\!48}{22\!\cdots\!01}a^{9}+\frac{65\!\cdots\!36}{22\!\cdots\!01}a^{8}+\frac{78\!\cdots\!31}{22\!\cdots\!01}a^{7}+\frac{34\!\cdots\!58}{22\!\cdots\!01}a^{6}+\frac{29\!\cdots\!13}{22\!\cdots\!01}a^{5}+\frac{99\!\cdots\!01}{22\!\cdots\!01}a^{4}+\frac{57\!\cdots\!24}{22\!\cdots\!01}a^{3}+\frac{16\!\cdots\!72}{22\!\cdots\!01}a^{2}+\frac{48\!\cdots\!78}{22\!\cdots\!01}a+\frac{30\!\cdots\!59}{52\!\cdots\!07}$, $\frac{49\!\cdots\!07}{10\!\cdots\!47}a^{11}-\frac{12\!\cdots\!41}{10\!\cdots\!47}a^{10}+\frac{48\!\cdots\!68}{10\!\cdots\!47}a^{9}-\frac{17\!\cdots\!30}{10\!\cdots\!47}a^{8}+\frac{31\!\cdots\!94}{10\!\cdots\!47}a^{7}-\frac{10\!\cdots\!93}{10\!\cdots\!47}a^{6}+\frac{13\!\cdots\!43}{10\!\cdots\!47}a^{5}-\frac{33\!\cdots\!68}{10\!\cdots\!47}a^{4}+\frac{27\!\cdots\!81}{10\!\cdots\!47}a^{3}-\frac{57\!\cdots\!72}{10\!\cdots\!47}a^{2}+\frac{22\!\cdots\!89}{10\!\cdots\!47}a-\frac{94\!\cdots\!64}{24\!\cdots\!29}$, $\frac{94\!\cdots\!73}{10\!\cdots\!47}a^{11}-\frac{18\!\cdots\!37}{10\!\cdots\!47}a^{10}+\frac{29\!\cdots\!74}{22\!\cdots\!01}a^{9}-\frac{27\!\cdots\!62}{10\!\cdots\!47}a^{8}+\frac{10\!\cdots\!01}{10\!\cdots\!47}a^{7}-\frac{17\!\cdots\!58}{10\!\cdots\!47}a^{6}+\frac{47\!\cdots\!55}{10\!\cdots\!47}a^{5}-\frac{57\!\cdots\!92}{10\!\cdots\!47}a^{4}+\frac{11\!\cdots\!36}{10\!\cdots\!47}a^{3}-\frac{97\!\cdots\!73}{10\!\cdots\!47}a^{2}+\frac{10\!\cdots\!12}{10\!\cdots\!47}a-\frac{14\!\cdots\!50}{24\!\cdots\!29}$, $\frac{15\!\cdots\!76}{10\!\cdots\!47}a^{11}+\frac{23\!\cdots\!28}{10\!\cdots\!47}a^{10}+\frac{51\!\cdots\!20}{22\!\cdots\!01}a^{9}+\frac{21\!\cdots\!79}{10\!\cdots\!47}a^{8}+\frac{15\!\cdots\!10}{10\!\cdots\!47}a^{7}+\frac{90\!\cdots\!08}{10\!\cdots\!47}a^{6}+\frac{55\!\cdots\!76}{10\!\cdots\!47}a^{5}+\frac{33\!\cdots\!59}{10\!\cdots\!47}a^{4}+\frac{10\!\cdots\!14}{10\!\cdots\!47}a^{3}+\frac{93\!\cdots\!10}{10\!\cdots\!47}a^{2}+\frac{89\!\cdots\!98}{10\!\cdots\!47}a+\frac{23\!\cdots\!75}{24\!\cdots\!29}$
| 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: | \( 1059.54542703 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1059.54542703 \cdot 14932}{2\cdot\sqrt{7787035719905674198640625}}\cr\approx \mathstrut & 0.174421771774 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 176*x^10 - 241*x^9 + 13764*x^8 - 20921*x^7 + 610782*x^6 - 868596*x^5 + 16022269*x^4 - 17677113*x^3 + 231278288*x^2 - 141901568*x + 1423267621)
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 + 176*x^10 - 241*x^9 + 13764*x^8 - 20921*x^7 + 610782*x^6 - 868596*x^5 + 16022269*x^4 - 17677113*x^3 + 231278288*x^2 - 141901568*x + 1423267621, 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 + 176*x^10 - 241*x^9 + 13764*x^8 - 20921*x^7 + 610782*x^6 - 868596*x^5 + 16022269*x^4 - 17677113*x^3 + 231278288*x^2 - 141901568*x + 1423267621);
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 + 176*x^10 - 241*x^9 + 13764*x^8 - 20921*x^7 + 610782*x^6 - 868596*x^5 + 16022269*x^4 - 17677113*x^3 + 231278288*x^2 - 141901568*x + 1423267621);
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
| \(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\), 4.0.1105425.1, 6.6.11796113.1 |
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 | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | R | R | R | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.1.0.1}{1} }^{12}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.12.6.1 | $x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |
|
\(5\)
| 5.12.6.2 | $x^{12} + 25 x^{8} - 500 x^{6} + 625 x^{4} + 31250$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |
|
\(7\)
| 7.12.8.1 | $x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |