Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 76*x^10 + 19*x^9 + 1915*x^8 + 834*x^7 - 18925*x^6 - 17455*x^5 + 60172*x^4 + 61645*x^3 - 50203*x^2 - 64242*x - 12836)
gp: K = bnfinit(y^12 - y^11 - 76*y^10 + 19*y^9 + 1915*y^8 + 834*y^7 - 18925*y^6 - 17455*y^5 + 60172*y^4 + 61645*y^3 - 50203*y^2 - 64242*y - 12836, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 76*x^10 + 19*x^9 + 1915*x^8 + 834*x^7 - 18925*x^6 - 17455*x^5 + 60172*x^4 + 61645*x^3 - 50203*x^2 - 64242*x - 12836);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 76*x^10 + 19*x^9 + 1915*x^8 + 834*x^7 - 18925*x^6 - 17455*x^5 + 60172*x^4 + 61645*x^3 - 50203*x^2 - 64242*x - 12836)
\( x^{12} - x^{11} - 76 x^{10} + 19 x^{9} + 1915 x^{8} + 834 x^{7} - 18925 x^{6} - 17455 x^{5} + \cdots - 12836 \)
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: |
\(1386078850992348503443697\)
\(\medspace = 17^{9}\cdot 43^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(102.76\) | 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: | $17^{3/4}43^{2/3}\approx 102.75800437868217$ | ||
| Ramified primes: |
\(17\), \(43\)
| 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: | \(731=17\cdot 43\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{731}(608,·)$, $\chi_{731}(1,·)$, $\chi_{731}(259,·)$, $\chi_{731}(135,·)$, $\chi_{731}(681,·)$, $\chi_{731}(302,·)$, $\chi_{731}(208,·)$, $\chi_{731}(307,·)$, $\chi_{731}(565,·)$, $\chi_{731}(560,·)$, $\chi_{731}(251,·)$, $\chi_{731}(509,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{4}+\frac{1}{8}a^{3}-\frac{3}{8}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{30\!\cdots\!32}a^{11}-\frac{25\!\cdots\!81}{76\!\cdots\!08}a^{10}+\frac{39\!\cdots\!41}{76\!\cdots\!08}a^{9}+\frac{28\!\cdots\!95}{30\!\cdots\!32}a^{8}+\frac{37\!\cdots\!59}{15\!\cdots\!16}a^{7}+\frac{923803706231274}{40\!\cdots\!41}a^{6}+\frac{59\!\cdots\!83}{30\!\cdots\!32}a^{5}+\frac{86\!\cdots\!13}{38\!\cdots\!54}a^{4}-\frac{69\!\cdots\!61}{19\!\cdots\!27}a^{3}+\frac{60\!\cdots\!89}{30\!\cdots\!32}a^{2}+\frac{40\!\cdots\!03}{15\!\cdots\!16}a+\frac{32\!\cdots\!33}{76\!\cdots\!08}$
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
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: |
$\frac{914801468352599}{30\!\cdots\!32}a^{11}-\frac{617255465454089}{76\!\cdots\!08}a^{10}-\frac{16\!\cdots\!99}{76\!\cdots\!08}a^{9}+\frac{12\!\cdots\!17}{30\!\cdots\!32}a^{8}+\frac{77\!\cdots\!49}{15\!\cdots\!16}a^{7}-\frac{21\!\cdots\!16}{40\!\cdots\!41}a^{6}-\frac{15\!\cdots\!07}{30\!\cdots\!32}a^{5}+\frac{34\!\cdots\!30}{19\!\cdots\!27}a^{4}+\frac{65\!\cdots\!91}{38\!\cdots\!54}a^{3}-\frac{43\!\cdots\!57}{30\!\cdots\!32}a^{2}-\frac{24\!\cdots\!95}{15\!\cdots\!16}a-\frac{33\!\cdots\!09}{76\!\cdots\!08}$, $\frac{467953297826681}{15\!\cdots\!16}a^{11}-\frac{245826561980369}{76\!\cdots\!08}a^{10}-\frac{17\!\cdots\!61}{76\!\cdots\!08}a^{9}+\frac{11\!\cdots\!91}{15\!\cdots\!16}a^{8}+\frac{43\!\cdots\!77}{76\!\cdots\!08}a^{7}+\frac{681570835977462}{40\!\cdots\!41}a^{6}-\frac{81\!\cdots\!29}{15\!\cdots\!16}a^{5}-\frac{28\!\cdots\!83}{76\!\cdots\!08}a^{4}+\frac{10\!\cdots\!67}{76\!\cdots\!08}a^{3}+\frac{11\!\cdots\!81}{15\!\cdots\!16}a^{2}-\frac{66\!\cdots\!29}{76\!\cdots\!08}a-\frac{86\!\cdots\!93}{38\!\cdots\!54}$, $\frac{393238776228869}{30\!\cdots\!32}a^{11}-\frac{61448452944364}{19\!\cdots\!27}a^{10}-\frac{35\!\cdots\!83}{38\!\cdots\!54}a^{9}+\frac{48\!\cdots\!11}{30\!\cdots\!32}a^{8}+\frac{35\!\cdots\!71}{15\!\cdots\!16}a^{7}-\frac{780789100945836}{40\!\cdots\!41}a^{6}-\frac{75\!\cdots\!37}{30\!\cdots\!32}a^{5}+\frac{23\!\cdots\!35}{76\!\cdots\!08}a^{4}+\frac{77\!\cdots\!71}{76\!\cdots\!08}a^{3}+\frac{74\!\cdots\!29}{30\!\cdots\!32}a^{2}-\frac{17\!\cdots\!01}{15\!\cdots\!16}a-\frac{41\!\cdots\!31}{76\!\cdots\!08}$, $\frac{17\!\cdots\!53}{30\!\cdots\!32}a^{11}+\frac{18\!\cdots\!89}{15\!\cdots\!16}a^{10}-\frac{80\!\cdots\!43}{15\!\cdots\!16}a^{9}-\frac{28\!\cdots\!31}{30\!\cdots\!32}a^{8}+\frac{24\!\cdots\!89}{15\!\cdots\!16}a^{7}+\frac{17\!\cdots\!41}{80\!\cdots\!82}a^{6}-\frac{59\!\cdots\!69}{30\!\cdots\!32}a^{5}-\frac{27\!\cdots\!95}{15\!\cdots\!16}a^{4}+\frac{12\!\cdots\!37}{15\!\cdots\!16}a^{3}+\frac{84\!\cdots\!75}{30\!\cdots\!32}a^{2}-\frac{12\!\cdots\!15}{15\!\cdots\!16}a-\frac{18\!\cdots\!13}{76\!\cdots\!08}$, $\frac{54\!\cdots\!59}{30\!\cdots\!32}a^{11}-\frac{12\!\cdots\!35}{15\!\cdots\!16}a^{10}-\frac{16\!\cdots\!05}{15\!\cdots\!16}a^{9}+\frac{13\!\cdots\!79}{30\!\cdots\!32}a^{8}+\frac{31\!\cdots\!51}{15\!\cdots\!16}a^{7}-\frac{52\!\cdots\!93}{80\!\cdots\!82}a^{6}-\frac{49\!\cdots\!15}{30\!\cdots\!32}a^{5}+\frac{45\!\cdots\!17}{15\!\cdots\!16}a^{4}+\frac{76\!\cdots\!95}{15\!\cdots\!16}a^{3}-\frac{91\!\cdots\!47}{30\!\cdots\!32}a^{2}-\frac{88\!\cdots\!65}{15\!\cdots\!16}a-\frac{12\!\cdots\!79}{76\!\cdots\!08}$, $\frac{18\!\cdots\!53}{15\!\cdots\!16}a^{11}-\frac{34\!\cdots\!67}{15\!\cdots\!16}a^{10}-\frac{13\!\cdots\!03}{15\!\cdots\!16}a^{9}+\frac{75\!\cdots\!05}{76\!\cdots\!08}a^{8}+\frac{86\!\cdots\!55}{38\!\cdots\!54}a^{7}-\frac{34\!\cdots\!99}{40\!\cdots\!41}a^{6}-\frac{34\!\cdots\!45}{15\!\cdots\!16}a^{5}-\frac{43\!\cdots\!21}{15\!\cdots\!16}a^{4}+\frac{11\!\cdots\!27}{15\!\cdots\!16}a^{3}+\frac{97\!\cdots\!79}{76\!\cdots\!08}a^{2}-\frac{13\!\cdots\!39}{19\!\cdots\!27}a-\frac{35\!\cdots\!48}{19\!\cdots\!27}$, $\frac{12\!\cdots\!11}{30\!\cdots\!32}a^{11}-\frac{11\!\cdots\!15}{15\!\cdots\!16}a^{10}-\frac{46\!\cdots\!69}{15\!\cdots\!16}a^{9}+\frac{10\!\cdots\!75}{30\!\cdots\!32}a^{8}+\frac{11\!\cdots\!15}{15\!\cdots\!16}a^{7}-\frac{26\!\cdots\!05}{80\!\cdots\!82}a^{6}-\frac{23\!\cdots\!03}{30\!\cdots\!32}a^{5}-\frac{11\!\cdots\!47}{15\!\cdots\!16}a^{4}+\frac{40\!\cdots\!99}{15\!\cdots\!16}a^{3}+\frac{11\!\cdots\!65}{30\!\cdots\!32}a^{2}-\frac{37\!\cdots\!69}{15\!\cdots\!16}a-\frac{49\!\cdots\!79}{76\!\cdots\!08}$, $\frac{27\!\cdots\!75}{30\!\cdots\!32}a^{11}-\frac{37\!\cdots\!50}{19\!\cdots\!27}a^{10}-\frac{12\!\cdots\!52}{19\!\cdots\!27}a^{9}+\frac{29\!\cdots\!25}{30\!\cdots\!32}a^{8}+\frac{25\!\cdots\!81}{15\!\cdots\!16}a^{7}-\frac{92\!\cdots\!45}{80\!\cdots\!82}a^{6}-\frac{49\!\cdots\!95}{30\!\cdots\!32}a^{5}+\frac{19\!\cdots\!79}{76\!\cdots\!08}a^{4}+\frac{40\!\cdots\!73}{76\!\cdots\!08}a^{3}-\frac{14\!\cdots\!17}{30\!\cdots\!32}a^{2}-\frac{64\!\cdots\!99}{15\!\cdots\!16}a-\frac{64\!\cdots\!49}{76\!\cdots\!08}$, $\frac{208962924329857}{30\!\cdots\!32}a^{11}+\frac{14\!\cdots\!01}{15\!\cdots\!16}a^{10}-\frac{11\!\cdots\!71}{15\!\cdots\!16}a^{9}-\frac{21\!\cdots\!11}{30\!\cdots\!32}a^{8}+\frac{32\!\cdots\!85}{15\!\cdots\!16}a^{7}+\frac{12\!\cdots\!55}{80\!\cdots\!82}a^{6}-\frac{42\!\cdots\!73}{30\!\cdots\!32}a^{5}-\frac{19\!\cdots\!11}{15\!\cdots\!16}a^{4}-\frac{99\!\cdots\!75}{15\!\cdots\!16}a^{3}+\frac{58\!\cdots\!11}{30\!\cdots\!32}a^{2}+\frac{25\!\cdots\!57}{15\!\cdots\!16}a+\frac{22\!\cdots\!35}{76\!\cdots\!08}$, $\frac{37\!\cdots\!09}{30\!\cdots\!32}a^{11}-\frac{29\!\cdots\!31}{76\!\cdots\!08}a^{10}-\frac{32\!\cdots\!89}{38\!\cdots\!54}a^{9}+\frac{62\!\cdots\!91}{30\!\cdots\!32}a^{8}+\frac{28\!\cdots\!79}{15\!\cdots\!16}a^{7}-\frac{23\!\cdots\!05}{80\!\cdots\!82}a^{6}-\frac{50\!\cdots\!13}{30\!\cdots\!32}a^{5}+\frac{21\!\cdots\!47}{19\!\cdots\!27}a^{4}+\frac{33\!\cdots\!33}{76\!\cdots\!08}a^{3}-\frac{28\!\cdots\!75}{30\!\cdots\!32}a^{2}-\frac{52\!\cdots\!61}{15\!\cdots\!16}a-\frac{59\!\cdots\!31}{76\!\cdots\!08}$, $\frac{112050222058335}{51\!\cdots\!12}a^{11}-\frac{4382864875337}{51\!\cdots\!12}a^{10}-\frac{83\!\cdots\!25}{51\!\cdots\!12}a^{9}-\frac{15\!\cdots\!31}{12\!\cdots\!78}a^{8}+\frac{24\!\cdots\!01}{649192901406239}a^{7}+\frac{15\!\cdots\!83}{27625229847074}a^{6}-\frac{15\!\cdots\!51}{51\!\cdots\!12}a^{5}-\frac{32\!\cdots\!79}{51\!\cdots\!12}a^{4}+\frac{15\!\cdots\!21}{51\!\cdots\!12}a^{3}+\frac{66\!\cdots\!78}{649192901406239}a^{2}+\frac{55\!\cdots\!29}{12\!\cdots\!78}a+\frac{24\!\cdots\!35}{649192901406239}$
| 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: | \( 441712046.214 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 441712046.214 \cdot 1}{2\cdot\sqrt{1386078850992348503443697}}\cr\approx \mathstrut & 0.768378540273 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 76*x^10 + 19*x^9 + 1915*x^8 + 834*x^7 - 18925*x^6 - 17455*x^5 + 60172*x^4 + 61645*x^3 - 50203*x^2 - 64242*x - 12836)
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 - 76*x^10 + 19*x^9 + 1915*x^8 + 834*x^7 - 18925*x^6 - 17455*x^5 + 60172*x^4 + 61645*x^3 - 50203*x^2 - 64242*x - 12836, 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 - 76*x^10 + 19*x^9 + 1915*x^8 + 834*x^7 - 18925*x^6 - 17455*x^5 + 60172*x^4 + 61645*x^3 - 50203*x^2 - 64242*x - 12836);
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 - 76*x^10 + 19*x^9 + 1915*x^8 + 834*x^7 - 18925*x^6 - 17455*x^5 + 60172*x^4 + 61645*x^3 - 50203*x^2 - 64242*x - 12836);
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
| \(\Q(\sqrt{17}) \), 3.3.1849.1, 4.4.4913.1, 6.6.16796569313.1 |
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 | ${\href{/padicField/2.2.0.1}{2} }^{6}$ | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/47.1.0.1}{1} }^{12}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(43\)
| 43.6.4.1 | $x^{6} + 126 x^{5} + 5301 x^{4} + 74930 x^{3} + 21321 x^{2} + 227916 x + 3171406$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 43.6.4.1 | $x^{6} + 126 x^{5} + 5301 x^{4} + 74930 x^{3} + 21321 x^{2} + 227916 x + 3171406$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |