Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 66*x^10 - 170*x^9 + 1249*x^8 - 3446*x^7 + 12689*x^6 - 57227*x^5 + 67939*x^4 - 83461*x^3 + 539865*x^2 - 552644*x + 152881)
gp: K = bnfinit(y^12 - y^11 + 66*y^10 - 170*y^9 + 1249*y^8 - 3446*y^7 + 12689*y^6 - 57227*y^5 + 67939*y^4 - 83461*y^3 + 539865*y^2 - 552644*y + 152881, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 66*x^10 - 170*x^9 + 1249*x^8 - 3446*x^7 + 12689*x^6 - 57227*x^5 + 67939*x^4 - 83461*x^3 + 539865*x^2 - 552644*x + 152881);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 + 66*x^10 - 170*x^9 + 1249*x^8 - 3446*x^7 + 12689*x^6 - 57227*x^5 + 67939*x^4 - 83461*x^3 + 539865*x^2 - 552644*x + 152881)
\( x^{12} - x^{11} + 66 x^{10} - 170 x^{9} + 1249 x^{8} - 3446 x^{7} + 12689 x^{6} - 57227 x^{5} + \cdots + 152881 \)
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: |
\(161428875495388931828125\)
\(\medspace = 5^{6}\cdot 7^{8}\cdot 13^{11}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(85.90\) | 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: | $5^{1/2}7^{2/3}13^{11/12}\approx 85.90104262646453$ | ||
| Ramified primes: |
\(5\), \(7\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(455=5\cdot 7\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{455}(256,·)$, $\chi_{455}(1,·)$, $\chi_{455}(99,·)$, $\chi_{455}(214,·)$, $\chi_{455}(296,·)$, $\chi_{455}(239,·)$, $\chi_{455}(16,·)$, $\chi_{455}(246,·)$, $\chi_{455}(184,·)$, $\chi_{455}(186,·)$, $\chi_{455}(219,·)$, $\chi_{455}(319,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | 4.0.54925.1$^{2}$, 12.0.161428875495388931828125.1$^{30}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{782}a^{9}-\frac{71}{782}a^{8}-\frac{4}{23}a^{7}+\frac{121}{782}a^{6}-\frac{5}{46}a^{5}+\frac{337}{782}a^{4}-\frac{12}{391}a^{3}+\frac{89}{782}a^{2}+\frac{227}{782}a$, $\frac{1}{782}a^{10}-\frac{47}{391}a^{8}+\frac{120}{391}a^{7}-\frac{48}{391}a^{6}+\frac{167}{782}a^{5}+\frac{26}{391}a^{4}+\frac{10}{23}a^{3}+\frac{145}{391}a^{2}+\frac{43}{391}a-\frac{1}{2}$, $\frac{1}{16\!\cdots\!86}a^{11}-\frac{44\!\cdots\!53}{99\!\cdots\!58}a^{10}+\frac{82\!\cdots\!88}{84\!\cdots\!93}a^{9}-\frac{86\!\cdots\!50}{84\!\cdots\!93}a^{8}+\frac{20\!\cdots\!61}{84\!\cdots\!93}a^{7}-\frac{82\!\cdots\!33}{16\!\cdots\!86}a^{6}+\frac{84\!\cdots\!87}{16\!\cdots\!86}a^{5}+\frac{42\!\cdots\!94}{84\!\cdots\!93}a^{4}+\frac{79\!\cdots\!65}{49\!\cdots\!29}a^{3}-\frac{41\!\cdots\!76}{84\!\cdots\!93}a^{2}-\frac{34\!\cdots\!05}{73\!\cdots\!82}a-\frac{21\!\cdots\!01}{18\!\cdots\!02}$
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
$C_{13}\times C_{78}$, which has order $1014$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $338$ (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: |
$\frac{553359443973002}{65\!\cdots\!57}a^{11}-\frac{32\!\cdots\!61}{65\!\cdots\!57}a^{10}+\frac{24\!\cdots\!78}{38\!\cdots\!21}a^{9}-\frac{15\!\cdots\!16}{38\!\cdots\!21}a^{8}+\frac{13\!\cdots\!46}{65\!\cdots\!57}a^{7}-\frac{53\!\cdots\!16}{65\!\cdots\!57}a^{6}+\frac{18\!\cdots\!08}{65\!\cdots\!57}a^{5}-\frac{64\!\cdots\!04}{65\!\cdots\!57}a^{4}+\frac{17\!\cdots\!06}{65\!\cdots\!57}a^{3}-\frac{25\!\cdots\!78}{65\!\cdots\!57}a^{2}+\frac{70\!\cdots\!38}{28\!\cdots\!59}a-\frac{59\!\cdots\!23}{16\!\cdots\!27}$, $\frac{73\!\cdots\!77}{16\!\cdots\!86}a^{11}+\frac{29\!\cdots\!47}{16\!\cdots\!86}a^{10}+\frac{30\!\cdots\!73}{16\!\cdots\!86}a^{9}+\frac{93\!\cdots\!09}{84\!\cdots\!93}a^{8}-\frac{33\!\cdots\!59}{16\!\cdots\!86}a^{7}+\frac{97\!\cdots\!90}{49\!\cdots\!29}a^{6}-\frac{74\!\cdots\!47}{16\!\cdots\!86}a^{5}+\frac{14\!\cdots\!14}{84\!\cdots\!93}a^{4}-\frac{14\!\cdots\!45}{16\!\cdots\!86}a^{3}+\frac{10\!\cdots\!55}{16\!\cdots\!86}a^{2}+\frac{10\!\cdots\!61}{73\!\cdots\!82}a+\frac{47\!\cdots\!29}{94\!\cdots\!01}$, $\frac{80\!\cdots\!87}{16\!\cdots\!86}a^{11}+\frac{19\!\cdots\!21}{16\!\cdots\!86}a^{10}+\frac{54\!\cdots\!61}{16\!\cdots\!86}a^{9}+\frac{23\!\cdots\!32}{84\!\cdots\!93}a^{8}+\frac{81\!\cdots\!47}{16\!\cdots\!86}a^{7}+\frac{16\!\cdots\!87}{84\!\cdots\!93}a^{6}+\frac{54\!\cdots\!47}{16\!\cdots\!86}a^{5}-\frac{57\!\cdots\!04}{49\!\cdots\!29}a^{4}-\frac{64\!\cdots\!35}{16\!\cdots\!86}a^{3}-\frac{79\!\cdots\!65}{16\!\cdots\!86}a^{2}+\frac{70\!\cdots\!75}{73\!\cdots\!82}a+\frac{31\!\cdots\!17}{94\!\cdots\!01}$, $\frac{33\!\cdots\!67}{99\!\cdots\!58}a^{11}+\frac{34\!\cdots\!37}{16\!\cdots\!86}a^{10}+\frac{19\!\cdots\!54}{84\!\cdots\!93}a^{9}+\frac{83\!\cdots\!41}{84\!\cdots\!93}a^{8}+\frac{17\!\cdots\!09}{84\!\cdots\!93}a^{7}+\frac{23\!\cdots\!61}{16\!\cdots\!86}a^{6}-\frac{16\!\cdots\!43}{16\!\cdots\!86}a^{5}+\frac{29\!\cdots\!68}{84\!\cdots\!93}a^{4}-\frac{69\!\cdots\!46}{84\!\cdots\!93}a^{3}+\frac{54\!\cdots\!89}{49\!\cdots\!29}a^{2}+\frac{46\!\cdots\!27}{73\!\cdots\!82}a+\frac{18\!\cdots\!05}{18\!\cdots\!02}$, $\frac{11\!\cdots\!21}{16\!\cdots\!86}a^{11}+\frac{12\!\cdots\!69}{16\!\cdots\!86}a^{10}+\frac{33\!\cdots\!63}{84\!\cdots\!93}a^{9}+\frac{17\!\cdots\!67}{84\!\cdots\!93}a^{8}+\frac{25\!\cdots\!19}{84\!\cdots\!93}a^{7}+\frac{17\!\cdots\!61}{16\!\cdots\!86}a^{6}-\frac{21\!\cdots\!53}{16\!\cdots\!86}a^{5}-\frac{13\!\cdots\!52}{84\!\cdots\!93}a^{4}-\frac{10\!\cdots\!38}{84\!\cdots\!93}a^{3}-\frac{22\!\cdots\!01}{84\!\cdots\!93}a^{2}+\frac{11\!\cdots\!41}{73\!\cdots\!82}a+\frac{11\!\cdots\!39}{18\!\cdots\!02}$
| 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: | \( 6176.96689838 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 6176.96689838 \cdot 1014}{2\cdot\sqrt{161428875495388931828125}}\cr\approx \mathstrut & 0.479591892500 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 66*x^10 - 170*x^9 + 1249*x^8 - 3446*x^7 + 12689*x^6 - 57227*x^5 + 67939*x^4 - 83461*x^3 + 539865*x^2 - 552644*x + 152881)
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 + 66*x^10 - 170*x^9 + 1249*x^8 - 3446*x^7 + 12689*x^6 - 57227*x^5 + 67939*x^4 - 83461*x^3 + 539865*x^2 - 552644*x + 152881, 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 + 66*x^10 - 170*x^9 + 1249*x^8 - 3446*x^7 + 12689*x^6 - 57227*x^5 + 67939*x^4 - 83461*x^3 + 539865*x^2 - 552644*x + 152881);
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 + 66*x^10 - 170*x^9 + 1249*x^8 - 3446*x^7 + 12689*x^6 - 57227*x^5 + 67939*x^4 - 83461*x^3 + 539865*x^2 - 552644*x + 152881);
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
| \(\Q(\sqrt{13}) \), 3.3.8281.1, 4.0.54925.1, 6.6.891474493.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{3}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | R | R | ${\href{/padicField/11.12.0.1}{12} }$ | R | ${\href{/padicField/17.1.0.1}{1} }^{12}$ | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.1.0.1}{1} }^{12}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.12.6.2 | $x^{12} + 25 x^{8} - 500 x^{6} + 625 x^{4} + 31250$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |
|
\(7\)
| 7.12.8.2 | $x^{12} - 70 x^{9} + 1519 x^{6} - 4802 x^{3} + 21609$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
|
\(13\)
| 13.12.11.1 | $x^{12} + 156$ | $12$ | $1$ | $11$ | $C_{12}$ | $[\ ]_{12}$ |