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Properties

Label 12.0.677298770454137.1
Degree $12$
Signature $[0, 6]$
Discriminant $6.773\times 10^{14}$
Root discriminant \(17.21\)
Ramified primes $13,17$
Class number $1$
Class group trivial
Galois group $D_4 \times C_3$ (as 12T14)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 2*x^10 - 5*x^9 + 16*x^8 + 7*x^7 + 26*x^6 - 3*x^5 + 108*x^4 + 183*x^3 + 294*x^2 + 184*x + 103)
 
gp: K = bnfinit(y^12 - y^11 - 2*y^10 - 5*y^9 + 16*y^8 + 7*y^7 + 26*y^6 - 3*y^5 + 108*y^4 + 183*y^3 + 294*y^2 + 184*y + 103, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 2*x^10 - 5*x^9 + 16*x^8 + 7*x^7 + 26*x^6 - 3*x^5 + 108*x^4 + 183*x^3 + 294*x^2 + 184*x + 103);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - x^11 - 2*x^10 - 5*x^9 + 16*x^8 + 7*x^7 + 26*x^6 - 3*x^5 + 108*x^4 + 183*x^3 + 294*x^2 + 184*x + 103)
 

\( x^{12} - x^{11} - 2 x^{10} - 5 x^{9} + 16 x^{8} + 7 x^{7} + 26 x^{6} - 3 x^{5} + 108 x^{4} + 183 x^{3} + \cdots + 103 \) 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:  $[0, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(677298770454137\) \(\medspace = 13^{10}\cdot 17^{3}\) 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:  \(17.21\)
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:  $13^{5/6}17^{1/2}\approx 34.95510311561147$
Ramified primes:   \(13\), \(17\) 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:  \(\Q(\sqrt{17}) \)
$\card{ \Aut(K/\Q) }$:  $6$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  4.0.3757.1$^{2}$, deg 12$^{6}$, deg 24$^{24}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{3}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{21256690353}a^{11}-\frac{2610891961}{21256690353}a^{10}+\frac{381460484}{21256690353}a^{9}-\frac{4973723846}{21256690353}a^{8}-\frac{4354777672}{21256690353}a^{7}-\frac{7107766376}{21256690353}a^{6}-\frac{741038569}{21256690353}a^{5}-\frac{2151446792}{7085563451}a^{4}-\frac{2366365883}{7085563451}a^{3}+\frac{283527575}{21256690353}a^{2}-\frac{1113594808}{7085563451}a+\frac{3391700304}{7085563451}$ 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:  $5$
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:   $\frac{466075348}{21256690353}a^{11}-\frac{875732101}{21256690353}a^{10}-\frac{201449946}{7085563451}a^{9}-\frac{286254630}{7085563451}a^{8}+\frac{8736545071}{21256690353}a^{7}-\frac{4399401283}{21256690353}a^{6}+\frac{8360908795}{21256690353}a^{5}-\frac{4605863345}{21256690353}a^{4}+\frac{51846263258}{21256690353}a^{3}+\frac{42926337533}{21256690353}a^{2}+\frac{15196806389}{7085563451}a-\frac{942194221}{21256690353}$, $\frac{644531036}{21256690353}a^{11}-\frac{1488767378}{21256690353}a^{10}-\frac{62503363}{7085563451}a^{9}-\frac{448524167}{7085563451}a^{8}+\frac{12763118159}{21256690353}a^{7}-\frac{9745216385}{21256690353}a^{6}+\frac{14938932434}{21256690353}a^{5}-\frac{16010448664}{21256690353}a^{4}+\frac{67415314306}{21256690353}a^{3}+\frac{42652901410}{21256690353}a^{2}+\frac{18980501128}{7085563451}a-\frac{29650999475}{21256690353}$, $\frac{17050574}{7085563451}a^{11}-\frac{194839010}{7085563451}a^{10}+\frac{1374484079}{21256690353}a^{9}-\frac{610091870}{21256690353}a^{8}+\frac{1013064145}{21256690353}a^{7}-\frac{8309443315}{21256690353}a^{6}+\frac{13183018450}{21256690353}a^{5}-\frac{14324096044}{21256690353}a^{4}+\frac{4528954450}{21256690353}a^{3}-\frac{9076079877}{7085563451}a^{2}-\frac{2001926372}{7085563451}a-\frac{5923842233}{21256690353}$, $\frac{737181247}{21256690353}a^{11}-\frac{1811425153}{21256690353}a^{10}+\frac{441217291}{21256690353}a^{9}-\frac{2245237027}{21256690353}a^{8}+\frac{4667368787}{7085563451}a^{7}-\frac{4464499629}{7085563451}a^{6}+\frac{7736738594}{7085563451}a^{5}-\frac{16075635757}{21256690353}a^{4}+\frac{81621075802}{21256690353}a^{3}+\frac{41902874453}{21256690353}a^{2}+\frac{20477375959}{7085563451}a+\frac{12473252629}{21256690353}$, $\frac{593379314}{21256690353}a^{11}-\frac{904250348}{21256690353}a^{10}-\frac{1561994168}{21256690353}a^{9}-\frac{735480631}{21256690353}a^{8}+\frac{11750054014}{21256690353}a^{7}-\frac{1435773070}{21256690353}a^{6}+\frac{1755913984}{21256690353}a^{5}-\frac{562117540}{7085563451}a^{4}+\frac{20962119952}{7085563451}a^{3}+\frac{69881141041}{21256690353}a^{2}+\frac{20982427500}{7085563451}a-\frac{7909052414}{7085563451}$ 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:  \( 120.784031363 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 120.784031363 \cdot 1}{2\cdot\sqrt{677298770454137}}\cr\approx \mathstrut & 0.142780399138 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 2*x^10 - 5*x^9 + 16*x^8 + 7*x^7 + 26*x^6 - 3*x^5 + 108*x^4 + 183*x^3 + 294*x^2 + 184*x + 103)
 
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 - 2*x^10 - 5*x^9 + 16*x^8 + 7*x^7 + 26*x^6 - 3*x^5 + 108*x^4 + 183*x^3 + 294*x^2 + 184*x + 103, 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 - 2*x^10 - 5*x^9 + 16*x^8 + 7*x^7 + 26*x^6 - 3*x^5 + 108*x^4 + 183*x^3 + 294*x^2 + 184*x + 103);
 
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 - 2*x^10 - 5*x^9 + 16*x^8 + 7*x^7 + 26*x^6 - 3*x^5 + 108*x^4 + 183*x^3 + 294*x^2 + 184*x + 103);
 
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

$C_3\times D_4$ (as 12T14):

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 15 conjugacy class representatives for $D_4 \times C_3$
Character table for $D_4 \times C_3$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, 4.0.2873.1, \(\Q(\zeta_{13})^+\)

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: deg 12
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{2}$ ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{3}$ ${\href{/padicField/7.12.0.1}{12} }$ ${\href{/padicField/11.12.0.1}{12} }$ R R ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{3}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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
\(13\) Copy content Toggle raw display 13.12.10.1$x^{12} + 72 x^{11} + 2172 x^{10} + 35280 x^{9} + 328380 x^{8} + 1703232 x^{7} + 4282170 x^{6} + 3407400 x^{5} + 1340820 x^{4} + 712800 x^{3} + 3855192 x^{2} + 18082080 x + 35650393$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
\(17\) Copy content Toggle raw display 17.3.0.1$x^{3} + x + 14$$1$$3$$0$$C_3$$[\ ]^{3}$
17.3.0.1$x^{3} + x + 14$$1$$3$$0$$C_3$$[\ ]^{3}$
17.6.3.1$x^{6} + 459 x^{5} + 70280 x^{4} + 3597823 x^{3} + 1271380 x^{2} + 4696159 x + 50437479$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$