Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 265*x^10 - 436*x^9 + 30354*x^8 - 40216*x^7 + 1919630*x^6 - 1949106*x^5 + 70585208*x^4 - 49564380*x^3 + 1429811640*x^2 - 529942832*x + 12452899201)
gp: K = bnfinit(y^12 - 2*y^11 + 265*y^10 - 436*y^9 + 30354*y^8 - 40216*y^7 + 1919630*y^6 - 1949106*y^5 + 70585208*y^4 - 49564380*y^3 + 1429811640*y^2 - 529942832*y + 12452899201, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 + 265*x^10 - 436*x^9 + 30354*x^8 - 40216*x^7 + 1919630*x^6 - 1949106*x^5 + 70585208*x^4 - 49564380*x^3 + 1429811640*x^2 - 529942832*x + 12452899201);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 + 265*x^10 - 436*x^9 + 30354*x^8 - 40216*x^7 + 1919630*x^6 - 1949106*x^5 + 70585208*x^4 - 49564380*x^3 + 1429811640*x^2 - 529942832*x + 12452899201)
\( x^{12} - 2 x^{11} + 265 x^{10} - 436 x^{9} + 30354 x^{8} - 40216 x^{7} + 1919630 x^{6} + \cdots + 12452899201 \)
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: |
\(7991252743075419172110336\)
\(\medspace = 2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 23^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(118.91\) | 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: | $2^{3/2}3^{1/2}7^{5/6}23^{1/2}\approx 118.9098703368596$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3864=2^{3}\cdot 3\cdot 7\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3864}(2945,·)$, $\chi_{3864}(2437,·)$, $\chi_{3864}(1885,·)$, $\chi_{3864}(1,·)$, $\chi_{3864}(2209,·)$, $\chi_{3864}(781,·)$, $\chi_{3864}(1517,·)$, $\chi_{3864}(3313,·)$, $\chi_{3864}(2393,·)$, $\chi_{3864}(185,·)$, $\chi_{3864}(2621,·)$, $\chi_{3864}(965,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-46}) \), \(\Q(\sqrt{-966}) \), 6.0.14957039104.1$^{3}$, 6.0.2826880390656.6$^{3}$, 12.0.7991252743075419172110336.16$^{24}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{14\!\cdots\!37}a^{11}-\frac{68\!\cdots\!80}{14\!\cdots\!37}a^{10}-\frac{64\!\cdots\!71}{16\!\cdots\!39}a^{9}+\frac{47\!\cdots\!02}{14\!\cdots\!37}a^{8}+\frac{46\!\cdots\!15}{14\!\cdots\!37}a^{7}+\frac{46\!\cdots\!23}{14\!\cdots\!37}a^{6}-\frac{16\!\cdots\!95}{10\!\cdots\!49}a^{5}-\frac{20\!\cdots\!10}{14\!\cdots\!37}a^{4}-\frac{29\!\cdots\!49}{14\!\cdots\!37}a^{3}-\frac{30\!\cdots\!23}{14\!\cdots\!37}a^{2}+\frac{19\!\cdots\!40}{14\!\cdots\!37}a+\frac{92\!\cdots\!06}{32\!\cdots\!59}$
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
$C_{10}\times C_{17640}$, which has order $176400$ (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: | $5$ | 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: |
$\frac{62\!\cdots\!90}{20\!\cdots\!29}a^{11}-\frac{12\!\cdots\!47}{20\!\cdots\!29}a^{10}+\frac{19\!\cdots\!10}{24\!\cdots\!63}a^{9}-\frac{27\!\cdots\!84}{20\!\cdots\!29}a^{8}+\frac{18\!\cdots\!40}{20\!\cdots\!29}a^{7}-\frac{25\!\cdots\!81}{20\!\cdots\!29}a^{6}+\frac{10\!\cdots\!22}{20\!\cdots\!29}a^{5}-\frac{12\!\cdots\!61}{20\!\cdots\!29}a^{4}+\frac{31\!\cdots\!10}{20\!\cdots\!29}a^{3}-\frac{31\!\cdots\!03}{20\!\cdots\!29}a^{2}+\frac{39\!\cdots\!66}{20\!\cdots\!29}a-\frac{77\!\cdots\!66}{47\!\cdots\!03}$, $\frac{62\!\cdots\!90}{20\!\cdots\!29}a^{11}-\frac{12\!\cdots\!47}{20\!\cdots\!29}a^{10}+\frac{19\!\cdots\!10}{24\!\cdots\!63}a^{9}-\frac{27\!\cdots\!84}{20\!\cdots\!29}a^{8}+\frac{18\!\cdots\!40}{20\!\cdots\!29}a^{7}-\frac{25\!\cdots\!81}{20\!\cdots\!29}a^{6}+\frac{10\!\cdots\!22}{20\!\cdots\!29}a^{5}-\frac{12\!\cdots\!61}{20\!\cdots\!29}a^{4}+\frac{31\!\cdots\!10}{20\!\cdots\!29}a^{3}-\frac{31\!\cdots\!03}{20\!\cdots\!29}a^{2}+\frac{39\!\cdots\!66}{20\!\cdots\!29}a-\frac{72\!\cdots\!63}{47\!\cdots\!03}$, $\frac{45\!\cdots\!26}{14\!\cdots\!37}a^{11}-\frac{89\!\cdots\!11}{14\!\cdots\!37}a^{10}+\frac{14\!\cdots\!88}{16\!\cdots\!39}a^{9}-\frac{19\!\cdots\!51}{14\!\cdots\!37}a^{8}+\frac{13\!\cdots\!64}{14\!\cdots\!37}a^{7}-\frac{18\!\cdots\!54}{14\!\cdots\!37}a^{6}+\frac{59\!\cdots\!22}{10\!\cdots\!49}a^{5}-\frac{87\!\cdots\!41}{14\!\cdots\!37}a^{4}+\frac{23\!\cdots\!64}{14\!\cdots\!37}a^{3}-\frac{22\!\cdots\!56}{14\!\cdots\!37}a^{2}+\frac{29\!\cdots\!32}{14\!\cdots\!37}a-\frac{52\!\cdots\!09}{32\!\cdots\!59}$, $\frac{36\!\cdots\!46}{14\!\cdots\!37}a^{11}-\frac{13\!\cdots\!45}{14\!\cdots\!37}a^{10}+\frac{92\!\cdots\!38}{16\!\cdots\!39}a^{9}-\frac{29\!\cdots\!04}{14\!\cdots\!37}a^{8}+\frac{68\!\cdots\!74}{14\!\cdots\!37}a^{7}-\frac{27\!\cdots\!86}{14\!\cdots\!37}a^{6}+\frac{24\!\cdots\!66}{10\!\cdots\!49}a^{5}-\frac{13\!\cdots\!43}{14\!\cdots\!37}a^{4}+\frac{77\!\cdots\!54}{14\!\cdots\!37}a^{3}-\frac{33\!\cdots\!77}{14\!\cdots\!37}a^{2}+\frac{79\!\cdots\!36}{14\!\cdots\!37}a-\frac{83\!\cdots\!66}{32\!\cdots\!59}$, $\frac{44\!\cdots\!54}{14\!\cdots\!37}a^{11}-\frac{73\!\cdots\!43}{14\!\cdots\!37}a^{10}+\frac{13\!\cdots\!28}{16\!\cdots\!39}a^{9}-\frac{14\!\cdots\!12}{14\!\cdots\!37}a^{8}+\frac{12\!\cdots\!90}{14\!\cdots\!37}a^{7}-\frac{12\!\cdots\!13}{14\!\cdots\!37}a^{6}+\frac{57\!\cdots\!94}{10\!\cdots\!49}a^{5}-\frac{41\!\cdots\!61}{14\!\cdots\!37}a^{4}+\frac{22\!\cdots\!96}{14\!\cdots\!37}a^{3}-\frac{28\!\cdots\!34}{14\!\cdots\!37}a^{2}+\frac{27\!\cdots\!62}{14\!\cdots\!37}a+\frac{24\!\cdots\!50}{32\!\cdots\!59}$
| 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: | \( 140.7987960054707 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 140.7987960054707 \cdot 176400}{2\cdot\sqrt{7991252743075419172110336}}\cr\approx \mathstrut & 0.270295803525798 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 265*x^10 - 436*x^9 + 30354*x^8 - 40216*x^7 + 1919630*x^6 - 1949106*x^5 + 70585208*x^4 - 49564380*x^3 + 1429811640*x^2 - 529942832*x + 12452899201)
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 - 2*x^11 + 265*x^10 - 436*x^9 + 30354*x^8 - 40216*x^7 + 1919630*x^6 - 1949106*x^5 + 70585208*x^4 - 49564380*x^3 + 1429811640*x^2 - 529942832*x + 12452899201, 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 - 2*x^11 + 265*x^10 - 436*x^9 + 30354*x^8 - 40216*x^7 + 1919630*x^6 - 1949106*x^5 + 70585208*x^4 - 49564380*x^3 + 1429811640*x^2 - 529942832*x + 12452899201);
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 - 2*x^11 + 265*x^10 - 436*x^9 + 30354*x^8 - 40216*x^7 + 1919630*x^6 - 1949106*x^5 + 70585208*x^4 - 49564380*x^3 + 1429811640*x^2 - 529942832*x + 12452899201);
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
$C_2\times C_6$ (as 12T2):
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)
| An abelian group of order 12 |
| The 12 conjugacy class representatives for $C_6\times C_2$ |
| Character table for $C_6\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{-46}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-966}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{21}, \sqrt{-46})\), 6.0.14957039104.1, \(\Q(\zeta_{21})^+\), 6.0.2826880390656.6 |
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 | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.1.0.1}{1} }^{12}$ | ${\href{/padicField/43.1.0.1}{1} }^{12}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.18.23 | $x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ |
|
\(3\)
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(7\)
| 7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |
|
\(23\)
| 23.12.6.1 | $x^{12} + 140 x^{10} + 18 x^{9} + 8092 x^{8} + 20 x^{7} + 244001 x^{6} - 56140 x^{5} + 4059563 x^{4} - 1721304 x^{3} + 36312468 x^{2} - 14694000 x + 141161628$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |