Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 16*x^10 - 24*x^9 + 111*x^8 - 101*x^7 + 566*x^6 - 314*x^5 + 1256*x^4 - 1635*x^3 + 851*x^2 - 2345*x + 1789)
gp: K = bnfinit(y^12 - 2*y^11 + 16*y^10 - 24*y^9 + 111*y^8 - 101*y^7 + 566*y^6 - 314*y^5 + 1256*y^4 - 1635*y^3 + 851*y^2 - 2345*y + 1789, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 + 16*x^10 - 24*x^9 + 111*x^8 - 101*x^7 + 566*x^6 - 314*x^5 + 1256*x^4 - 1635*x^3 + 851*x^2 - 2345*x + 1789);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 2*x^11 + 16*x^10 - 24*x^9 + 111*x^8 - 101*x^7 + 566*x^6 - 314*x^5 + 1256*x^4 - 1635*x^3 + 851*x^2 - 2345*x + 1789)
\( x^{12} - 2 x^{11} + 16 x^{10} - 24 x^{9} + 111 x^{8} - 101 x^{7} + 566 x^{6} - 314 x^{5} + 1256 x^{4} + \cdots + 1789 \)
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: |
\(4032054328696714817\)
\(\medspace = 7^{6}\cdot 17^{11}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(35.52\) | 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: | $7^{1/2}17^{11/12}\approx 35.51898373246793$ | ||
| Ramified primes: |
\(7\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{94}a^{10}+\frac{17}{94}a^{9}+\frac{33}{94}a^{8}-\frac{20}{47}a^{7}-\frac{31}{94}a^{6}+\frac{35}{94}a^{5}-\frac{23}{47}a^{4}+\frac{20}{47}a^{3}-\frac{29}{94}a^{2}+\frac{3}{94}a-\frac{41}{94}$, $\frac{1}{66118799882434}a^{11}+\frac{79844269959}{66118799882434}a^{10}-\frac{14687626133893}{66118799882434}a^{9}-\frac{1695760126149}{33059399941217}a^{8}+\frac{11390047555729}{66118799882434}a^{7}+\frac{1225984077111}{66118799882434}a^{6}-\frac{133161608911}{703391488111}a^{5}+\frac{8720567002306}{33059399941217}a^{4}-\frac{21948503719349}{66118799882434}a^{3}-\frac{19961313729457}{66118799882434}a^{2}-\frac{13910542351241}{66118799882434}a-\frac{4338546507884}{33059399941217}$
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
$C_{2}$, which has order $2$
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: |
$\frac{80952492386}{33059399941217}a^{11}-\frac{208857866008}{33059399941217}a^{10}+\frac{1234905674494}{33059399941217}a^{9}-\frac{2054164918212}{33059399941217}a^{8}+\frac{6927568016449}{33059399941217}a^{7}-\frac{5130371448060}{33059399941217}a^{6}+\frac{26690714021444}{33059399941217}a^{5}-\frac{11416386768611}{33059399941217}a^{4}+\frac{16446562695653}{33059399941217}a^{3}-\frac{45502732270935}{33059399941217}a^{2}-\frac{60681033881803}{33059399941217}a+\frac{68270214840082}{33059399941217}$, $\frac{82188154973}{33059399941217}a^{11}-\frac{454010707349}{66118799882434}a^{10}+\frac{3441786095821}{66118799882434}a^{9}-\frac{6553930142341}{66118799882434}a^{8}+\frac{13804176238654}{33059399941217}a^{7}-\frac{29137760740591}{66118799882434}a^{6}+\frac{123648767999893}{66118799882434}a^{5}-\frac{20985873082578}{33059399941217}a^{4}+\frac{152147219613430}{33059399941217}a^{3}-\frac{24197818423997}{66118799882434}a^{2}+\frac{304691185667805}{66118799882434}a-\frac{81224376219629}{66118799882434}$, $\frac{22494529761}{33059399941217}a^{11}-\frac{35238372457}{33059399941217}a^{10}+\frac{331249283889}{33059399941217}a^{9}-\frac{307752982672}{33059399941217}a^{8}+\frac{43761708139}{703391488111}a^{7}+\frac{80749521998}{33059399941217}a^{6}+\frac{8430740045419}{33059399941217}a^{5}+\frac{3918811431560}{33059399941217}a^{4}+\frac{10507145394930}{33059399941217}a^{3}-\frac{4080073232305}{33059399941217}a^{2}-\frac{6630377589530}{33059399941217}a-\frac{5375784398871}{33059399941217}$, $\frac{25987226358}{33059399941217}a^{11}-\frac{7601562776}{33059399941217}a^{10}+\frac{397308297962}{33059399941217}a^{9}-\frac{88520136970}{33059399941217}a^{8}+\frac{2902861825405}{33059399941217}a^{7}+\frac{1058150964340}{33059399941217}a^{6}+\frac{17431324865724}{33059399941217}a^{5}+\frac{17916288419223}{33059399941217}a^{4}+\frac{57721290105845}{33059399941217}a^{3}+\frac{18255952431283}{33059399941217}a^{2}+\frac{16788631626709}{33059399941217}a-\frac{41527970827294}{33059399941217}$, $\frac{763366073589}{66118799882434}a^{11}-\frac{997913728099}{66118799882434}a^{10}+\frac{11949950727355}{66118799882434}a^{9}-\frac{5483836053640}{33059399941217}a^{8}+\frac{83767977621051}{66118799882434}a^{7}-\frac{28825633214727}{66118799882434}a^{6}+\frac{226023844270570}{33059399941217}a^{5}+\frac{24407267609783}{33059399941217}a^{4}+\frac{11\!\cdots\!59}{66118799882434}a^{3}-\frac{479373089985925}{66118799882434}a^{2}+\frac{559599366419321}{66118799882434}a-\frac{852160484839126}{33059399941217}$
| 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: | \( 7218.657057374844 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 7218.657057374844 \cdot 2}{2\cdot\sqrt{4032054328696714817}}\cr\approx \mathstrut & 0.221193536578973 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 16*x^10 - 24*x^9 + 111*x^8 - 101*x^7 + 566*x^6 - 314*x^5 + 1256*x^4 - 1635*x^3 + 851*x^2 - 2345*x + 1789)
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 + 16*x^10 - 24*x^9 + 111*x^8 - 101*x^7 + 566*x^6 - 314*x^5 + 1256*x^4 - 1635*x^3 + 851*x^2 - 2345*x + 1789, 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 + 16*x^10 - 24*x^9 + 111*x^8 - 101*x^7 + 566*x^6 - 314*x^5 + 1256*x^4 - 1635*x^3 + 851*x^2 - 2345*x + 1789);
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 + 16*x^10 - 24*x^9 + 111*x^8 - 101*x^7 + 566*x^6 - 314*x^5 + 1256*x^4 - 1635*x^3 + 851*x^2 - 2345*x + 1789);
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
$C_4\times S_3$ (as 12T11):
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 solvable group of order 24 |
| The 12 conjugacy class representatives for $S_3 \times C_4$ |
| Character table for $S_3 \times C_4$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 3.1.2023.1, 4.0.240737.1, 6.2.69572993.1 |
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]
Sibling fields
| Galois closure: | deg 24 |
| Degree 12 sibling: | 12.4.82286823034626833.1 |
| Minimal sibling: | 12.4.82286823034626833.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.4.2.2 | $x^{4} - 42 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(17\)
| 17.12.11.2 | $x^{12} + 34$ | $12$ | $1$ | $11$ | $S_3 \times C_4$ | $[\ ]_{12}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.119.2t1.a.a | $1$ | $ 7 \cdot 17 $ | \(\Q(\sqrt{-119}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.17.4t1.a.a | $1$ | $ 17 $ | 4.4.4913.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
| * | 1.119.4t1.a.a | $1$ | $ 7 \cdot 17 $ | 4.0.240737.1 | $C_4$ (as 4T1) | $0$ | $-1$ |
| 1.17.4t1.a.b | $1$ | $ 17 $ | 4.4.4913.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
| * | 1.119.4t1.a.b | $1$ | $ 7 \cdot 17 $ | 4.0.240737.1 | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 2.2023.3t2.b.a | $2$ | $ 7 \cdot 17^{2}$ | 3.1.2023.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.2023.6t3.b.a | $2$ | $ 7 \cdot 17^{2}$ | 6.0.487010951.1 | $D_{6}$ (as 6T3) | $1$ | $0$ |
| * | 2.2023.12t11.b.a | $2$ | $ 7 \cdot 17^{2}$ | 12.0.4032054328696714817.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ |
| * | 2.2023.12t11.b.b | $2$ | $ 7 \cdot 17^{2}$ | 12.0.4032054328696714817.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.