Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 33*x^10 - 112*x^9 + 1575*x^8 + 7056*x^7 - 3355*x^6 - 165312*x^5 - 167664*x^4 + 2470720*x^3 + 17419956*x^2 + 42136416*x + 63660176)
gp: K = bnfinit(y^12 - 33*y^10 - 112*y^9 + 1575*y^8 + 7056*y^7 - 3355*y^6 - 165312*y^5 - 167664*y^4 + 2470720*y^3 + 17419956*y^2 + 42136416*y + 63660176, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 33*x^10 - 112*x^9 + 1575*x^8 + 7056*x^7 - 3355*x^6 - 165312*x^5 - 167664*x^4 + 2470720*x^3 + 17419956*x^2 + 42136416*x + 63660176);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 33*x^10 - 112*x^9 + 1575*x^8 + 7056*x^7 - 3355*x^6 - 165312*x^5 - 167664*x^4 + 2470720*x^3 + 17419956*x^2 + 42136416*x + 63660176)
\( x^{12} - 33 x^{10} - 112 x^{9} + 1575 x^{8} + 7056 x^{7} - 3355 x^{6} - 165312 x^{5} - 167664 x^{4} + \cdots + 63660176 \)
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: |
\(120538181723674419070377725952\)
\(\medspace = 2^{12}\cdot 3^{16}\cdot 7^{8}\cdot 17^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(265.11\) | 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: | $2\cdot 3^{4/3}7^{2/3}17^{3/4}\approx 265.1105758191317$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(17\)
| 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: | \(4284=2^{2}\cdot 3^{2}\cdot 7\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4284}(1,·)$, $\chi_{4284}(3523,·)$, $\chi_{4284}(1633,·)$, $\chi_{4284}(3931,·)$, $\chi_{4284}(3175,·)$, $\chi_{4284}(781,·)$, $\chi_{4284}(1135,·)$, $\chi_{4284}(3025,·)$, $\chi_{4284}(1891,·)$, $\chi_{4284}(373,·)$, $\chi_{4284}(2041,·)$, $\chi_{4284}(2767,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | 4.0.78608.1$^{2}$, 12.0.120538181723674419070377725952.1$^{30}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{5873723792}a^{10}-\frac{233455923}{5873723792}a^{9}-\frac{10192011}{734215474}a^{8}-\frac{37325801}{2936861896}a^{7}+\frac{468162947}{5873723792}a^{6}-\frac{1153734393}{5873723792}a^{5}+\frac{240721173}{1468430948}a^{4}-\frac{1406278617}{2936861896}a^{3}-\frac{7797835}{367107737}a^{2}-\frac{129611785}{367107737}a+\frac{143991879}{367107737}$, $\frac{1}{19\!\cdots\!44}a^{11}-\frac{1451912615}{19\!\cdots\!44}a^{10}+\frac{15\!\cdots\!65}{97\!\cdots\!72}a^{9}+\frac{18\!\cdots\!77}{48\!\cdots\!36}a^{8}+\frac{27\!\cdots\!71}{19\!\cdots\!44}a^{7}-\frac{15\!\cdots\!17}{19\!\cdots\!44}a^{6}-\frac{93\!\cdots\!37}{97\!\cdots\!72}a^{5}+\frac{68\!\cdots\!45}{48\!\cdots\!36}a^{4}-\frac{30\!\cdots\!17}{24\!\cdots\!18}a^{3}+\frac{29\!\cdots\!75}{48\!\cdots\!36}a^{2}+\frac{34\!\cdots\!26}{12\!\cdots\!59}a-\frac{32\!\cdots\!33}{12\!\cdots\!59}$
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
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2886}$, which has order $92352$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $30784$ (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: |
$\frac{378}{367107737}a^{11}-\frac{879}{367107737}a^{10}-\frac{14868}{367107737}a^{9}+\frac{30495}{734215474}a^{8}+\frac{689346}{367107737}a^{7}+\frac{659278}{367107737}a^{6}-\frac{10771068}{367107737}a^{5}-\frac{66162717}{734215474}a^{4}+\frac{143972836}{367107737}a^{3}+\frac{916447368}{367107737}a^{2}+\frac{2329201392}{367107737}a-\frac{1156605213}{367107737}$, $\frac{6087693192507}{97\!\cdots\!72}a^{11}-\frac{19797830158331}{48\!\cdots\!36}a^{10}-\frac{60910953190077}{97\!\cdots\!72}a^{9}+\frac{98912752513203}{24\!\cdots\!18}a^{8}+\frac{86\!\cdots\!03}{97\!\cdots\!72}a^{7}-\frac{10\!\cdots\!01}{48\!\cdots\!36}a^{6}-\frac{49\!\cdots\!35}{97\!\cdots\!72}a^{5}-\frac{52\!\cdots\!51}{12\!\cdots\!59}a^{4}+\frac{12\!\cdots\!29}{48\!\cdots\!36}a^{3}+\frac{17\!\cdots\!31}{24\!\cdots\!18}a^{2}+\frac{29\!\cdots\!55}{12\!\cdots\!59}a-\frac{50\!\cdots\!11}{12\!\cdots\!59}$, $\frac{894798904581}{48\!\cdots\!36}a^{11}-\frac{2100995845137}{24\!\cdots\!18}a^{10}-\frac{10219230505147}{48\!\cdots\!36}a^{9}+\frac{6627997962999}{24\!\cdots\!18}a^{8}+\frac{11\!\cdots\!01}{48\!\cdots\!36}a^{7}-\frac{428543998912029}{24\!\cdots\!18}a^{6}+\frac{14\!\cdots\!67}{48\!\cdots\!36}a^{5}-\frac{24\!\cdots\!51}{24\!\cdots\!18}a^{4}+\frac{10\!\cdots\!43}{24\!\cdots\!18}a^{3}+\frac{73\!\cdots\!29}{12\!\cdots\!59}a^{2}+\frac{40\!\cdots\!90}{12\!\cdots\!59}a-\frac{37\!\cdots\!45}{12\!\cdots\!59}$, $\frac{11\!\cdots\!39}{48\!\cdots\!36}a^{11}-\frac{31\!\cdots\!59}{48\!\cdots\!36}a^{10}-\frac{41\!\cdots\!81}{48\!\cdots\!36}a^{9}+\frac{17\!\cdots\!81}{24\!\cdots\!18}a^{8}+\frac{21\!\cdots\!45}{48\!\cdots\!36}a^{7}+\frac{12\!\cdots\!45}{48\!\cdots\!36}a^{6}-\frac{33\!\cdots\!81}{48\!\cdots\!36}a^{5}-\frac{24\!\cdots\!11}{12\!\cdots\!59}a^{4}+\frac{25\!\cdots\!87}{24\!\cdots\!18}a^{3}+\frac{61\!\cdots\!04}{12\!\cdots\!59}a^{2}+\frac{14\!\cdots\!42}{12\!\cdots\!59}a+\frac{23\!\cdots\!97}{12\!\cdots\!59}$, $\frac{312621058760037}{48\!\cdots\!36}a^{11}-\frac{15\!\cdots\!13}{24\!\cdots\!18}a^{10}-\frac{33\!\cdots\!27}{48\!\cdots\!36}a^{9}+\frac{23\!\cdots\!70}{12\!\cdots\!59}a^{8}+\frac{99\!\cdots\!49}{48\!\cdots\!36}a^{7}-\frac{14\!\cdots\!67}{24\!\cdots\!18}a^{6}-\frac{32\!\cdots\!89}{48\!\cdots\!36}a^{5}-\frac{26\!\cdots\!93}{12\!\cdots\!59}a^{4}+\frac{28\!\cdots\!75}{24\!\cdots\!18}a^{3}+\frac{56\!\cdots\!73}{12\!\cdots\!59}a^{2}+\frac{16\!\cdots\!34}{12\!\cdots\!59}a+\frac{30\!\cdots\!93}{12\!\cdots\!59}$
| 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: | \( 1065625.9379812907 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1065625.9379812907 \cdot 92352}{2\cdot\sqrt{120538181723674419070377725952}}\cr\approx \mathstrut & 8.72043169080323 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 33*x^10 - 112*x^9 + 1575*x^8 + 7056*x^7 - 3355*x^6 - 165312*x^5 - 167664*x^4 + 2470720*x^3 + 17419956*x^2 + 42136416*x + 63660176)
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 - 33*x^10 - 112*x^9 + 1575*x^8 + 7056*x^7 - 3355*x^6 - 165312*x^5 - 167664*x^4 + 2470720*x^3 + 17419956*x^2 + 42136416*x + 63660176, 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 - 33*x^10 - 112*x^9 + 1575*x^8 + 7056*x^7 - 3355*x^6 - 165312*x^5 - 167664*x^4 + 2470720*x^3 + 17419956*x^2 + 42136416*x + 63660176);
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 - 33*x^10 - 112*x^9 + 1575*x^8 + 7056*x^7 - 3355*x^6 - 165312*x^5 - 167664*x^4 + 2470720*x^3 + 17419956*x^2 + 42136416*x + 63660176);
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
| \(\Q(\sqrt{17}) \), 3.3.3969.1, 4.0.78608.1, 6.6.77394297393.1 |
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 | R | R | ${\href{/padicField/5.12.0.1}{12} }$ | R | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.1.0.1}{1} }^{12}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
|
\(3\)
| 3.12.16.40 | $x^{12} + 24 x^{11} + 216 x^{10} + 840 x^{9} + 864 x^{8} - 2592 x^{7} - 4482 x^{6} + 8424 x^{5} + 25272 x^{4} + 4968 x^{3} + 29808 x^{2} + 139077$ | $3$ | $4$ | $16$ | $C_{12}$ | $[2]^{4}$ |
|
\(7\)
| 7.12.8.3 | $x^{12} + 245 x^{6} - 1372 x^{3} + 7203$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |