↑

Properties

Label 12.8.1363454074150441.1
Degree $12$
Signature $[8, 2]$
Discriminant $1.363\times 10^{15}$
Root discriminant \(18.25\)
Ramified primes $7,13$
Class number $1$
Class group trivial
Galois group $C_2^2 \times A_4$ (as 12T25)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 12*x^10 + 11*x^9 + 40*x^8 - 57*x^7 - 36*x^6 + 197*x^5 + 82*x^4 - 221*x^3 - 124*x^2 + 78*x + 43)
 
gp: K = bnfinit(y^12 - y^11 - 12*y^10 + 11*y^9 + 40*y^8 - 57*y^7 - 36*y^6 + 197*y^5 + 82*y^4 - 221*y^3 - 124*y^2 + 78*y + 43, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 12*x^10 + 11*x^9 + 40*x^8 - 57*x^7 - 36*x^6 + 197*x^5 + 82*x^4 - 221*x^3 - 124*x^2 + 78*x + 43);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - x^11 - 12*x^10 + 11*x^9 + 40*x^8 - 57*x^7 - 36*x^6 + 197*x^5 + 82*x^4 - 221*x^3 - 124*x^2 + 78*x + 43)
 

\( x^{12} - x^{11} - 12 x^{10} + 11 x^{9} + 40 x^{8} - 57 x^{7} - 36 x^{6} + 197 x^{5} + 82 x^{4} + \cdots + 43 \) Copy content Toggle raw display

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:  $[8, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1363454074150441\) \(\medspace = 7^{10}\cdot 13^{6}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(18.25\)
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:  $7^{5/6}13^{1/2}\approx 18.248200448594663$
Ramified primes:   \(7\), \(13\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13}a^{10}+\frac{5}{13}a^{9}-\frac{4}{13}a^{8}-\frac{6}{13}a^{7}+\frac{1}{13}a^{6}+\frac{3}{13}a^{5}-\frac{1}{13}a^{4}-\frac{5}{13}a^{3}-\frac{4}{13}a^{2}-\frac{5}{13}a-\frac{1}{13}$, $\frac{1}{586296607}a^{11}+\frac{19418004}{586296607}a^{10}+\frac{230765819}{586296607}a^{9}-\frac{102836830}{586296607}a^{8}-\frac{154848776}{586296607}a^{7}-\frac{1498530}{586296607}a^{6}-\frac{156529625}{586296607}a^{5}-\frac{54670221}{586296607}a^{4}-\frac{284713019}{586296607}a^{3}-\frac{195433554}{586296607}a^{2}+\frac{76544114}{586296607}a+\frac{290843160}{586296607}$ Copy content Toggle raw display

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 $1$

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:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $\frac{23129214}{586296607}a^{11}-\frac{10177521}{586296607}a^{10}-\frac{284718695}{586296607}a^{9}+\frac{111253496}{586296607}a^{8}+\frac{982715122}{586296607}a^{7}-\frac{942232354}{586296607}a^{6}-\frac{1156581473}{586296607}a^{5}+\frac{4285378555}{586296607}a^{4}+\frac{3321718330}{586296607}a^{3}-\frac{2940351498}{586296607}a^{2}-\frac{2785496031}{586296607}a+\frac{528634896}{586296607}$, $\frac{23129214}{586296607}a^{11}-\frac{10177521}{586296607}a^{10}-\frac{284718695}{586296607}a^{9}+\frac{111253496}{586296607}a^{8}+\frac{982715122}{586296607}a^{7}-\frac{942232354}{586296607}a^{6}-\frac{1156581473}{586296607}a^{5}+\frac{4285378555}{586296607}a^{4}+\frac{3321718330}{586296607}a^{3}-\frac{2940351498}{586296607}a^{2}-\frac{2785496031}{586296607}a-\frac{57661711}{586296607}$, $\frac{16093321}{586296607}a^{11}-\frac{24055965}{586296607}a^{10}-\frac{163767520}{586296607}a^{9}+\frac{220495030}{586296607}a^{8}+\frac{390157338}{586296607}a^{7}-\frac{772279006}{586296607}a^{6}-\frac{85795032}{586296607}a^{5}+\frac{2346490430}{586296607}a^{4}+\frac{917948348}{586296607}a^{3}-\frac{2600449723}{586296607}a^{2}-\frac{1214806896}{586296607}a+\frac{210363058}{586296607}$, $\frac{16093321}{586296607}a^{11}-\frac{24055965}{586296607}a^{10}-\frac{163767520}{586296607}a^{9}+\frac{220495030}{586296607}a^{8}+\frac{390157338}{586296607}a^{7}-\frac{772279006}{586296607}a^{6}-\frac{85795032}{586296607}a^{5}+\frac{2346490430}{586296607}a^{4}+\frac{917948348}{586296607}a^{3}-\frac{2600449723}{586296607}a^{2}-\frac{628510289}{586296607}a+\frac{796659665}{586296607}$, $\frac{7035893}{586296607}a^{11}+\frac{13878444}{586296607}a^{10}-\frac{120951175}{586296607}a^{9}-\frac{109241534}{586296607}a^{8}+\frac{45581368}{45099739}a^{7}-\frac{169953348}{586296607}a^{6}-\frac{1070786441}{586296607}a^{5}+\frac{1938888125}{586296607}a^{4}+\frac{2403769982}{586296607}a^{3}-\frac{339901775}{586296607}a^{2}-\frac{1570689135}{586296607}a-\frac{268024769}{586296607}$, $\frac{30965319}{586296607}a^{11}-\frac{50357811}{586296607}a^{10}-\frac{340848088}{586296607}a^{9}+\frac{561666959}{586296607}a^{8}+\frac{898636982}{586296607}a^{7}-\frac{184985094}{45099739}a^{6}+\frac{362253318}{586296607}a^{5}+\frac{6106204608}{586296607}a^{4}-\frac{1552525744}{586296607}a^{3}-\frac{6496594139}{586296607}a^{2}+\frac{634411266}{586296607}a+\frac{3032628623}{586296607}$, $\frac{55767971}{586296607}a^{11}-\frac{2604731}{586296607}a^{10}-\frac{713104631}{586296607}a^{9}+\frac{25281901}{586296607}a^{8}+\frac{2617906774}{586296607}a^{7}-\frac{1559307539}{586296607}a^{6}-\frac{3862218471}{586296607}a^{5}+\frac{9926143642}{586296607}a^{4}+\frac{914965262}{45099739}a^{3}-\frac{5595377989}{586296607}a^{2}-\frac{10357903741}{586296607}a-\frac{2120056061}{586296607}$, $\frac{3163092}{586296607}a^{11}+\frac{4461919}{586296607}a^{10}-\frac{32095799}{586296607}a^{9}-\frac{80843209}{586296607}a^{8}+\frac{100461252}{586296607}a^{7}+\frac{23847101}{45099739}a^{6}-\frac{402729492}{586296607}a^{5}-\frac{17241374}{586296607}a^{4}+\frac{1702820749}{586296607}a^{3}+\frac{640425531}{586296607}a^{2}-\frac{1423183622}{586296607}a-\frac{872855135}{586296607}$, $\frac{14998537}{586296607}a^{11}-\frac{15473888}{586296607}a^{10}-\frac{164419674}{586296607}a^{9}+\frac{144333968}{586296607}a^{8}+\frac{447517493}{586296607}a^{7}-\frac{663517872}{586296607}a^{6}-\frac{246892634}{586296607}a^{5}+\frac{2340592099}{586296607}a^{4}+\frac{1447595764}{586296607}a^{3}-\frac{1120666683}{586296607}a^{2}-\frac{1112069762}{586296607}a-\frac{162308573}{586296607}$ Copy content Toggle raw display
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:  \( 1604.61145419 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{2}\cdot 1604.61145419 \cdot 1}{2\cdot\sqrt{1363454074150441}}\cr\approx \mathstrut & 0.219593427484 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 12*x^10 + 11*x^9 + 40*x^8 - 57*x^7 - 36*x^6 + 197*x^5 + 82*x^4 - 221*x^3 - 124*x^2 + 78*x + 43)
 
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 - 12*x^10 + 11*x^9 + 40*x^8 - 57*x^7 - 36*x^6 + 197*x^5 + 82*x^4 - 221*x^3 - 124*x^2 + 78*x + 43, 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 - 12*x^10 + 11*x^9 + 40*x^8 - 57*x^7 - 36*x^6 + 197*x^5 + 82*x^4 - 221*x^3 - 124*x^2 + 78*x + 43);
 
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 - 12*x^10 + 11*x^9 + 40*x^8 - 57*x^7 - 36*x^6 + 197*x^5 + 82*x^4 - 221*x^3 - 124*x^2 + 78*x + 43);
 
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

$C_2^2\times A_4$ (as 12T25):

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 48
The 16 conjugacy class representatives for $C_2^2 \times A_4$
Character table for $C_2^2 \times A_4$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{7})^+\), 6.6.5274997.1, 6.4.218491.1, 6.4.2840383.1

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

Degree 12 siblings: 12.4.8067775586689.1, 12.0.47738317081.1, 12.4.8067775586689.2, 12.0.1363454074150441.2
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Minimal sibling: 12.0.47738317081.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{2}$ ${\href{/padicField/3.6.0.1}{6} }^{2}$ ${\href{/padicField/5.6.0.1}{6} }^{2}$ R ${\href{/padicField/11.6.0.1}{6} }^{2}$ R ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.3.0.1}{3} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.3.0.1}{3} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(7\) Copy content Toggle raw display 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}$
\(13\) Copy content Toggle raw display 13.2.1.1$x^{2} + 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} + 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} + 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} + 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$