Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 57*x^10 + 226*x^9 + 906*x^8 - 3428*x^7 - 5159*x^6 + 18006*x^5 + 10833*x^4 - 34220*x^3 - 4958*x^2 + 20914*x - 3371)
gp: K = bnfinit(y^12 - 4*y^11 - 57*y^10 + 226*y^9 + 906*y^8 - 3428*y^7 - 5159*y^6 + 18006*y^5 + 10833*y^4 - 34220*y^3 - 4958*y^2 + 20914*y - 3371, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 57*x^10 + 226*x^9 + 906*x^8 - 3428*x^7 - 5159*x^6 + 18006*x^5 + 10833*x^4 - 34220*x^3 - 4958*x^2 + 20914*x - 3371);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 - 57*x^10 + 226*x^9 + 906*x^8 - 3428*x^7 - 5159*x^6 + 18006*x^5 + 10833*x^4 - 34220*x^3 - 4958*x^2 + 20914*x - 3371)
\( x^{12} - 4 x^{11} - 57 x^{10} + 226 x^{9} + 906 x^{8} - 3428 x^{7} - 5159 x^{6} + 18006 x^{5} + \cdots - 3371 \)
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: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(737698544105842578608128\)
\(\medspace = 2^{12}\cdot 13^{9}\cdot 19^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(97.50\) | 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: | $2\cdot 13^{3/4}19^{2/3}\approx 97.49669871530843$ | ||
| Ramified primes: |
\(2\), \(13\), \(19\)
| 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: | \(988=2^{2}\cdot 13\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{988}(1,·)$, $\chi_{988}(805,·)$, $\chi_{988}(961,·)$, $\chi_{988}(619,·)$, $\chi_{988}(77,·)$, $\chi_{988}(463,·)$, $\chi_{988}(83,·)$, $\chi_{988}(885,·)$, $\chi_{988}(723,·)$, $\chi_{988}(343,·)$, $\chi_{988}(729,·)$, $\chi_{988}(239,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{645501}a^{10}+\frac{73201}{645501}a^{9}+\frac{32779}{215167}a^{8}+\frac{20743}{215167}a^{7}-\frac{5292}{215167}a^{6}-\frac{255293}{645501}a^{5}-\frac{57275}{215167}a^{4}-\frac{69142}{215167}a^{3}+\frac{307207}{645501}a^{2}-\frac{180713}{645501}a+\frac{225803}{645501}$, $\frac{1}{14337442826841}a^{11}-\frac{7656815}{14337442826841}a^{10}+\frac{441404369723}{4779147608947}a^{9}-\frac{1401821421569}{14337442826841}a^{8}-\frac{96516588300}{4779147608947}a^{7}-\frac{1657275285320}{14337442826841}a^{6}+\frac{16745023499}{4779147608947}a^{5}+\frac{3107172893549}{14337442826841}a^{4}-\frac{3662257903795}{14337442826841}a^{3}+\frac{5062562272355}{14337442826841}a^{2}+\frac{1814702873909}{4779147608947}a+\frac{1504049512114}{4779147608947}$
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
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: | $11$ | 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: |
$\frac{2090364732}{4779147608947}a^{11}-\frac{3064094550}{4779147608947}a^{10}-\frac{139389370792}{4779147608947}a^{9}+\frac{174503031618}{4779147608947}a^{8}+\frac{3005023414536}{4779147608947}a^{7}-\frac{2576592590670}{4779147608947}a^{6}-\frac{26352852012630}{4779147608947}a^{5}+\frac{13024930337517}{4779147608947}a^{4}+\frac{89534830049210}{4779147608947}a^{3}-\frac{23046622414719}{4779147608947}a^{2}-\frac{82937068675536}{4779147608947}a+\frac{15961309882959}{4779147608947}$, $\frac{1221704656}{4779147608947}a^{11}-\frac{3856674500}{4779147608947}a^{10}-\frac{75654465484}{4779147608947}a^{9}+\frac{224517601213}{4779147608947}a^{8}+\frac{1436977514832}{4779147608947}a^{7}-\frac{3621606614554}{4779147608947}a^{6}-\frac{10973664396788}{4779147608947}a^{5}+\frac{20913754151853}{4779147608947}a^{4}+\frac{33622527537640}{4779147608947}a^{3}-\frac{40820504142676}{4779147608947}a^{2}-\frac{29673244707528}{4779147608947}a+\frac{23088434754151}{4779147608947}$, $\frac{532}{645501}a^{11}-\frac{1996}{645501}a^{10}-\frac{30232}{645501}a^{9}+\frac{110161}{645501}a^{8}+\frac{159308}{215167}a^{7}-\frac{1564918}{645501}a^{6}-\frac{2710418}{645501}a^{5}+\frac{6964621}{645501}a^{4}+\frac{5879602}{645501}a^{3}-\frac{8587763}{645501}a^{2}-\frac{3404932}{645501}a+\frac{1028494}{645501}$, $\frac{2028655600}{4779147608947}a^{11}-\frac{31117474466}{14337442826841}a^{10}-\frac{311809041049}{14337442826841}a^{9}+\frac{569808939520}{4779147608947}a^{8}+\frac{1208181123280}{4779147608947}a^{7}-\frac{8040750199635}{4779147608947}a^{6}-\frac{2066725297097}{4779147608947}a^{5}+\frac{103116500347240}{14337442826841}a^{4}-\frac{26414986826783}{14337442826841}a^{3}-\frac{95679509054206}{14337442826841}a^{2}+\frac{30999625381549}{14337442826841}a-\frac{4565112859756}{14337442826841}$, $\frac{18087527608}{14337442826841}a^{11}-\frac{53526120286}{14337442826841}a^{10}-\frac{1089661373488}{14337442826841}a^{9}+\frac{2970332630755}{14337442826841}a^{8}+\frac{6543467726564}{4779147608947}a^{7}-\frac{42488705107048}{14337442826841}a^{6}-\frac{139260574488428}{14337442826841}a^{5}+\frac{193768362979312}{14337442826841}a^{4}+\frac{399198335113912}{14337442826841}a^{3}-\frac{259885599664340}{14337442826841}a^{2}-\frac{338776754587261}{14337442826841}a+\frac{85065603426172}{14337442826841}$, $\frac{9210453184}{14337442826841}a^{11}-\frac{46711576486}{14337442826841}a^{10}-\frac{480288545188}{14337442826841}a^{9}+\frac{2596867244686}{14337442826841}a^{8}+\frac{1970398412324}{4779147608947}a^{7}-\frac{37893969406690}{14337442826841}a^{6}-\frac{14064455603012}{14337442826841}a^{5}+\frac{178360043409769}{14337442826841}a^{4}-\frac{37143062568428}{14337442826841}a^{3}-\frac{244067377604054}{14337442826841}a^{2}+\frac{84163366170212}{14337442826841}a+\frac{58563048390871}{14337442826841}$, $\frac{18087527608}{14337442826841}a^{11}-\frac{53526120286}{14337442826841}a^{10}-\frac{1089661373488}{14337442826841}a^{9}+\frac{2970332630755}{14337442826841}a^{8}+\frac{6543467726564}{4779147608947}a^{7}-\frac{42488705107048}{14337442826841}a^{6}-\frac{139260574488428}{14337442826841}a^{5}+\frac{193768362979312}{14337442826841}a^{4}+\frac{399198335113912}{14337442826841}a^{3}-\frac{259885599664340}{14337442826841}a^{2}-\frac{324439311760420}{14337442826841}a+\frac{113740489079854}{14337442826841}$, $\frac{7097802133}{14337442826841}a^{11}-\frac{10201058147}{14337442826841}a^{10}-\frac{158084259148}{4779147608947}a^{9}+\frac{495076061200}{14337442826841}a^{8}+\frac{3552599124836}{4779147608947}a^{7}-\frac{4943019662237}{14337442826841}a^{6}-\frac{35900875527735}{4779147608947}a^{5}+\frac{21654648030194}{14337442826841}a^{4}+\frac{422875717422086}{14337442826841}a^{3}-\frac{59201395876513}{14337442826841}a^{2}-\frac{158633858256350}{4779147608947}a+\frac{36584402348744}{4779147608947}$, $\frac{1021031987}{14337442826841}a^{11}-\frac{6489865024}{14337442826841}a^{10}-\frac{18185748521}{4779147608947}a^{9}+\frac{121089446721}{4779147608947}a^{8}+\frac{225993326489}{4779147608947}a^{7}-\frac{1767609005736}{4779147608947}a^{6}-\frac{857537097362}{14337442826841}a^{5}+\frac{21775998353059}{14337442826841}a^{4}-\frac{3248294236569}{4779147608947}a^{3}-\frac{19791168895477}{14337442826841}a^{2}+\frac{12617499427742}{14337442826841}a+\frac{416621411276}{14337442826841}$, $\frac{58398190795}{14337442826841}a^{11}-\frac{319699344019}{14337442826841}a^{10}-\frac{2922932347045}{14337442826841}a^{9}+\frac{17714523452468}{14337442826841}a^{8}+\frac{30644580476576}{14337442826841}a^{7}-\frac{256219182276833}{14337442826841}a^{6}+\frac{4151931287823}{4779147608947}a^{5}+\frac{11\!\cdots\!52}{14337442826841}a^{4}-\frac{739895445751354}{14337442826841}a^{3}-\frac{13\!\cdots\!28}{14337442826841}a^{2}+\frac{13\!\cdots\!61}{14337442826841}a-\frac{213354330040193}{14337442826841}$, $\frac{25187009245}{4779147608947}a^{11}-\frac{90043361190}{4779147608947}a^{10}-\frac{4417447077490}{14337442826841}a^{9}+\frac{5015923658927}{4779147608947}a^{8}+\frac{74982802589299}{14337442826841}a^{7}-\frac{218791579664561}{14337442826841}a^{6}-\frac{496906935667483}{14337442826841}a^{5}+\frac{341319550524704}{4779147608947}a^{4}+\frac{13\!\cdots\!98}{14337442826841}a^{3}-\frac{436542169599973}{4779147608947}a^{2}-\frac{11\!\cdots\!26}{14337442826841}a+\frac{79647700595407}{4779147608947}$
| 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: | \( 59483998.235 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 59483998.235 \cdot 1}{2\cdot\sqrt{737698544105842578608128}}\cr\approx \mathstrut & 0.14183736065 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 57*x^10 + 226*x^9 + 906*x^8 - 3428*x^7 - 5159*x^6 + 18006*x^5 + 10833*x^4 - 34220*x^3 - 4958*x^2 + 20914*x - 3371)
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 - 4*x^11 - 57*x^10 + 226*x^9 + 906*x^8 - 3428*x^7 - 5159*x^6 + 18006*x^5 + 10833*x^4 - 34220*x^3 - 4958*x^2 + 20914*x - 3371, 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 - 4*x^11 - 57*x^10 + 226*x^9 + 906*x^8 - 3428*x^7 - 5159*x^6 + 18006*x^5 + 10833*x^4 - 34220*x^3 - 4958*x^2 + 20914*x - 3371);
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 - 4*x^11 - 57*x^10 + 226*x^9 + 906*x^8 - 3428*x^7 - 5159*x^6 + 18006*x^5 + 10833*x^4 - 34220*x^3 - 4958*x^2 + 20914*x - 3371);
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
| \(\Q(\sqrt{13}) \), 3.3.361.1, 4.4.35152.1, 6.6.286315237.1 |
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 | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.12.25 | $x^{12} + 12 x^{11} + 60 x^{10} + 160 x^{9} + 308 x^{8} + 736 x^{7} + 2272 x^{6} + 5632 x^{5} + 10608 x^{4} + 15040 x^{3} + 12224 x^{2} + 3584 x + 704$ | $2$ | $6$ | $12$ | $C_{12}$ | $[2]^{6}$ |
|
\(13\)
| 13.12.9.2 | $x^{12} + 8 x^{10} + 44 x^{9} + 63 x^{8} + 264 x^{7} + 550 x^{6} - 6336 x^{5} + 3843 x^{4} + 4532 x^{3} + 46454 x^{2} + 30668 x + 30982$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(19\)
| 19.12.8.1 | $x^{12} + 386 x^{10} + 109 x^{9} + 55308 x^{8} + 21792 x^{7} + 3500499 x^{6} + 2034936 x^{5} + 84821873 x^{4} + 99877907 x^{3} + 174885148 x^{2} + 920938017 x + 335157671$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |