Properties

Label 24.0.103...625.1
Degree $24$
Signature $[0, 12]$
Discriminant $1.035\times 10^{31}$
Root discriminant \(19.60\)
Ramified primes $3,5,7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - x^22 + 4*x^21 - 4*x^20 - 4*x^19 + 17*x^18 + 12*x^17 - 46*x^16 + 43*x^15 + 44*x^14 - 188*x^13 + 189*x^12 + 188*x^11 + 44*x^10 - 43*x^9 - 46*x^8 - 12*x^7 + 17*x^6 + 4*x^5 - 4*x^4 - 4*x^3 - x^2 + x + 1)
 
gp: K = bnfinit(y^24 - y^23 - y^22 + 4*y^21 - 4*y^20 - 4*y^19 + 17*y^18 + 12*y^17 - 46*y^16 + 43*y^15 + 44*y^14 - 188*y^13 + 189*y^12 + 188*y^11 + 44*y^10 - 43*y^9 - 46*y^8 - 12*y^7 + 17*y^6 + 4*y^5 - 4*y^4 - 4*y^3 - y^2 + y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - x^23 - x^22 + 4*x^21 - 4*x^20 - 4*x^19 + 17*x^18 + 12*x^17 - 46*x^16 + 43*x^15 + 44*x^14 - 188*x^13 + 189*x^12 + 188*x^11 + 44*x^10 - 43*x^9 - 46*x^8 - 12*x^7 + 17*x^6 + 4*x^5 - 4*x^4 - 4*x^3 - x^2 + x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - x^23 - x^22 + 4*x^21 - 4*x^20 - 4*x^19 + 17*x^18 + 12*x^17 - 46*x^16 + 43*x^15 + 44*x^14 - 188*x^13 + 189*x^12 + 188*x^11 + 44*x^10 - 43*x^9 - 46*x^8 - 12*x^7 + 17*x^6 + 4*x^5 - 4*x^4 - 4*x^3 - x^2 + x + 1)
 

\( x^{24} - x^{23} - x^{22} + 4 x^{21} - 4 x^{20} - 4 x^{19} + 17 x^{18} + 12 x^{17} - 46 x^{16} + 43 x^{15} + 44 x^{14} - 188 x^{13} + 189 x^{12} + 188 x^{11} + 44 x^{10} - 43 x^{9} - 46 x^{8} - 12 x^{7} + \cdots + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $24$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(10352754344108696148301025390625\) \(\medspace = 3^{12}\cdot 5^{12}\cdot 7^{20}\) 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:  \(19.60\)
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:  $3^{1/2}5^{1/2}7^{5/6}\approx 19.601711648537535$
Ramified primes:   \(3\), \(5\), \(7\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $24$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(105=3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{105}(64,·)$, $\chi_{105}(1,·)$, $\chi_{105}(4,·)$, $\chi_{105}(71,·)$, $\chi_{105}(74,·)$, $\chi_{105}(11,·)$, $\chi_{105}(76,·)$, $\chi_{105}(79,·)$, $\chi_{105}(16,·)$, $\chi_{105}(19,·)$, $\chi_{105}(86,·)$, $\chi_{105}(89,·)$, $\chi_{105}(26,·)$, $\chi_{105}(29,·)$, $\chi_{105}(94,·)$, $\chi_{105}(31,·)$, $\chi_{105}(34,·)$, $\chi_{105}(101,·)$, $\chi_{105}(104,·)$, $\chi_{105}(41,·)$, $\chi_{105}(44,·)$, $\chi_{105}(46,·)$, $\chi_{105}(59,·)$, $\chi_{105}(61,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13}a^{14}-\frac{5}{13}a^{7}-\frac{1}{13}$, $\frac{1}{13}a^{15}-\frac{5}{13}a^{8}-\frac{1}{13}a$, $\frac{1}{13}a^{16}-\frac{5}{13}a^{9}-\frac{1}{13}a^{2}$, $\frac{1}{13}a^{17}-\frac{5}{13}a^{10}-\frac{1}{13}a^{3}$, $\frac{1}{26}a^{18}-\frac{1}{26}a^{15}-\frac{1}{2}a^{12}+\frac{4}{13}a^{11}-\frac{1}{2}a^{9}-\frac{4}{13}a^{8}-\frac{1}{2}a^{6}+\frac{6}{13}a^{4}-\frac{1}{2}a^{3}-\frac{6}{13}a-\frac{1}{2}$, $\frac{1}{10946}a^{19}+\frac{50}{5473}a^{18}+\frac{159}{5473}a^{17}+\frac{25}{842}a^{16}-\frac{10}{5473}a^{15}-\frac{206}{5473}a^{14}-\frac{159}{842}a^{13}+\frac{1694}{5473}a^{12}+\frac{1440}{5473}a^{11}+\frac{477}{10946}a^{10}-\frac{192}{421}a^{9}-\frac{340}{5473}a^{8}-\frac{2685}{10946}a^{7}+\frac{10}{421}a^{6}+\frac{2606}{5473}a^{5}+\frac{1525}{10946}a^{4}-\frac{1134}{5473}a^{3}+\frac{128}{421}a^{2}-\frac{3321}{10946}a+\frac{1909}{5473}$, $\frac{1}{10946}a^{20}+\frac{1}{10946}a^{18}-\frac{321}{10946}a^{17}+\frac{159}{5473}a^{16}+\frac{25}{842}a^{15}+\frac{401}{10946}a^{14}+\frac{1057}{5473}a^{13}-\frac{159}{842}a^{12}-\frac{2085}{10946}a^{11}-\frac{244}{5473}a^{10}+\frac{477}{10946}a^{9}+\frac{37}{842}a^{8}+\frac{1344}{5473}a^{7}-\frac{4369}{10946}a^{6}-\frac{401}{842}a^{5}+\frac{2606}{5473}a^{4}-\frac{3527}{10946}a^{3}+\frac{3205}{10946}a^{2}+\frac{128}{421}a+\frac{1731}{10946}$, $\frac{1}{10946}a^{21}+\frac{4181}{10946}$, $\frac{1}{10946}a^{22}+\frac{4181}{10946}a$, $\frac{1}{10946}a^{23}+\frac{4181}{10946}a^{2}$ 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 $1$ (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:   \( -\frac{6765}{10946} a^{23} + \frac{6765}{10946} a^{22} + \frac{6765}{10946} a^{21} - \frac{27061}{10946} a^{20} + \frac{13530}{5473} a^{19} + \frac{13530}{5473} a^{18} - \frac{115005}{10946} a^{17} - \frac{40590}{5473} a^{16} + \frac{155595}{5473} a^{15} - \frac{290895}{10946} a^{14} - \frac{148830}{5473} a^{13} + \frac{635910}{5473} a^{12} - \frac{1278585}{10946} a^{11} - \frac{635910}{5473} a^{10} - \frac{148830}{5473} a^{9} + \frac{290895}{10946} a^{8} + \frac{155595}{5473} a^{7} + \frac{40590}{5473} a^{6} - \frac{115005}{10946} a^{5} - \frac{13530}{5473} a^{4} + \frac{13530}{5473} a^{3} + \frac{13530}{5473} a^{2} + \frac{6765}{10946} a - \frac{6765}{10946} \)  (order $42$) 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:   $\frac{5}{10946}a^{23}-\frac{5}{10946}a^{22}-\frac{5}{10946}a^{21}-\frac{125}{10946}a^{20}-\frac{10}{5473}a^{19}-\frac{10}{5473}a^{18}+\frac{85}{10946}a^{17}+\frac{30}{5473}a^{16}-\frac{115}{5473}a^{15}+\frac{215}{10946}a^{14}-\frac{1995}{5473}a^{13}-\frac{470}{5473}a^{12}+\frac{945}{10946}a^{11}+\frac{470}{5473}a^{10}+\frac{110}{5473}a^{9}-\frac{215}{10946}a^{8}-\frac{115}{5473}a^{7}-\frac{60654}{5473}a^{6}+\frac{85}{10946}a^{5}+\frac{10}{5473}a^{4}-\frac{10}{5473}a^{3}-\frac{10}{5473}a^{2}-\frac{5}{10946}a+\frac{5}{10946}$, $\frac{493}{5473}a^{23}+\frac{34}{13}a^{16}+\frac{987}{13}a^{9}+\frac{17}{5473}a^{2}-1$, $\frac{1291}{5473}a^{23}+\frac{1595}{10946}a^{22}-\frac{3190}{5473}a^{21}+\frac{3190}{5473}a^{20}+\frac{3190}{5473}a^{19}-\frac{27115}{10946}a^{18}+\frac{1045}{421}a^{17}+\frac{3923}{421}a^{16}-\frac{68585}{10946}a^{15}-\frac{35090}{5473}a^{14}+\frac{149930}{5473}a^{13}-\frac{301455}{10946}a^{12}-\frac{149930}{5473}a^{11}+\frac{49019}{421}a^{10}+\frac{69203}{842}a^{9}+\frac{36685}{5473}a^{8}+\frac{9570}{5473}a^{7}-\frac{27115}{10946}a^{6}-\frac{3190}{5473}a^{5}+\frac{3190}{5473}a^{4}+\frac{495}{842}a^{3}-\frac{13514}{5473}a^{2}-\frac{1595}{10946}a-\frac{1595}{10946}$, $\frac{233}{10946}a^{20}+\frac{145}{10946}a^{19}-\frac{1}{5473}a^{18}-\frac{55}{10946}a^{17}-\frac{29}{10946}a^{16}-\frac{4}{5473}a^{15}+\frac{1}{842}a^{14}+\frac{6765}{10946}a^{13}+\frac{2088}{5473}a^{12}-\frac{3}{10946}a^{11}-\frac{1597}{10946}a^{10}-\frac{493}{5473}a^{9}+\frac{1}{10946}a^{8}+\frac{29}{842}a^{7}+\frac{98209}{5473}a^{6}+\frac{121249}{10946}a^{5}-\frac{89}{10946}a^{4}-\frac{23184}{5473}a^{3}-\frac{28623}{10946}a^{2}+\frac{21}{10946}a$, $\frac{1597}{10946}a^{23}+\frac{1597}{10946}a^{22}-\frac{3194}{5473}a^{21}+\frac{3194}{5473}a^{20}+\frac{3194}{5473}a^{19}-\frac{27149}{10946}a^{18}+\frac{13573}{5473}a^{17}+\frac{36731}{5473}a^{16}-\frac{68671}{10946}a^{15}-\frac{35134}{5473}a^{14}+\frac{150118}{5473}a^{13}-\frac{301833}{10946}a^{12}-\frac{150118}{5473}a^{11}+\frac{637203}{5473}a^{10}+\frac{68671}{10946}a^{9}+\frac{36731}{5473}a^{8}+\frac{9582}{5473}a^{7}-\frac{27149}{10946}a^{6}-\frac{3194}{5473}a^{5}+\frac{3194}{5473}a^{4}-\frac{39925}{10946}a^{3}+\frac{1597}{10946}a^{2}-\frac{1597}{10946}a+\frac{9349}{10946}$, $\frac{5}{10946}a^{23}-\frac{5}{5473}a^{22}+\frac{25}{10946}a^{20}+\frac{105}{10946}a^{19}+\frac{105}{10946}a^{17}-\frac{25}{10946}a^{16}-\frac{145}{5473}a^{15}+\frac{445}{10946}a^{14}+\frac{5}{10946}a^{13}+\frac{1525}{5473}a^{12}+\frac{145}{842}a^{11}-\frac{5}{10946}a^{10}-\frac{360}{5473}a^{9}-\frac{435}{10946}a^{8}-\frac{15}{10946}a^{7}+\frac{85}{5473}a^{6}+\frac{121393}{10946}a^{5}-\frac{5}{842}a^{4}-\frac{20}{5473}a^{3}+\frac{15}{10946}a+\frac{5478}{5473}$, $\frac{987}{10946}a^{23}+\frac{1219}{10946}a^{22}-\frac{4181}{10946}a^{21}+\frac{1974}{5473}a^{20}+\frac{1974}{5473}a^{19}-\frac{16779}{10946}a^{18}+\frac{8392}{5473}a^{17}+\frac{22701}{5473}a^{16}-\frac{35705}{10946}a^{15}-\frac{25082}{5473}a^{14}+\frac{92778}{5473}a^{13}-\frac{186543}{10946}a^{12}-\frac{92778}{5473}a^{11}+\frac{393813}{5473}a^{10}+\frac{42441}{10946}a^{9}+\frac{120794}{5473}a^{8}-\frac{92171}{5473}a^{7}-\frac{16779}{10946}a^{6}-\frac{1974}{5473}a^{5}+\frac{1974}{5473}a^{4}-\frac{24675}{10946}a^{3}+\frac{11933}{10946}a^{2}-\frac{979}{10946}a+\frac{2885}{5473}$, $\frac{2583}{10946}a^{23}-\frac{2580}{5473}a^{22}-\frac{4}{5473}a^{21}+\frac{12913}{10946}a^{20}-\frac{20729}{10946}a^{19}-\frac{17}{5473}a^{18}+\frac{54177}{10946}a^{17}-\frac{12813}{10946}a^{16}-\frac{74892}{5473}a^{15}+\frac{229621}{10946}a^{14}+\frac{2957}{10946}a^{13}-\frac{300848}{5473}a^{12}+\frac{972661}{10946}a^{11}-\frac{2669}{10946}a^{10}-\frac{185789}{5473}a^{9}-\frac{224455}{10946}a^{8}-\frac{7719}{10946}a^{7}+\frac{43860}{5473}a^{6}-\frac{97}{10946}a^{5}-\frac{33545}{10946}a^{4}-\frac{33504}{5473}a^{3}+\frac{1}{5473}a^{2}+\frac{7741}{10946}a+\frac{8053}{5473}$, $\frac{1601}{10946}a^{23}+\frac{23}{421}a^{22}+\frac{381}{10946}a^{21}-\frac{106}{5473}a^{20}-\frac{22}{5473}a^{19}+\frac{19}{10946}a^{18}+\frac{86}{5473}a^{17}+\frac{23134}{5473}a^{16}+\frac{17443}{10946}a^{15}+\frac{5711}{5473}a^{14}-\frac{3474}{5473}a^{13}-\frac{937}{10946}a^{12}+\frac{1038}{5473}a^{11}+\frac{818}{5473}a^{10}+\frac{1344897}{10946}a^{9}+\frac{256569}{5473}a^{8}+\frac{158515}{5473}a^{7}-\frac{196231}{10946}a^{6}+\frac{74}{5473}a^{5}+\frac{10}{5473}a^{4}+\frac{74981}{10946}a^{3}-\frac{17691}{10946}a^{2}+\frac{8}{5473}a-\frac{5461}{5473}$, $\frac{6157}{10946}a^{23}-\frac{7}{10946}a^{22}-\frac{10713}{10946}a^{21}+\frac{9052}{5473}a^{20}-\frac{2375}{10946}a^{19}-\frac{43839}{10946}a^{18}+\frac{38794}{5473}a^{17}+\frac{168115}{10946}a^{16}-\frac{14327}{842}a^{15}+\frac{683}{421}a^{14}+\frac{480635}{10946}a^{13}-\frac{857045}{10946}a^{12}+\frac{59996}{5473}a^{11}+\frac{2062027}{10946}a^{10}+\frac{1542819}{10946}a^{9}+\frac{18949}{421}a^{8}-\frac{22431}{842}a^{7}-\frac{185713}{10946}a^{6}+\frac{55573}{5473}a^{5}+\frac{64561}{10946}a^{4}-\frac{31087}{10946}a^{3}-\frac{2003}{842}a^{2}-\frac{7737}{5473}a-\frac{5165}{5473}$, $\frac{3876}{5473}a^{23}-\frac{1597}{10946}a^{22}-\frac{5168}{5473}a^{21}+\frac{22879}{10946}a^{20}-\frac{5285}{5473}a^{19}-\frac{43929}{10946}a^{18}+\frac{98283}{10946}a^{17}+\frac{90470}{5473}a^{16}-\frac{232361}{10946}a^{15}+\frac{8943}{842}a^{14}+\frac{238715}{5473}a^{13}-\frac{1094621}{10946}a^{12}+\frac{488381}{10946}a^{11}+\frac{1031016}{5473}a^{10}+\frac{1916339}{10946}a^{9}+\frac{725109}{10946}a^{8}+\frac{871}{421}a^{7}-\frac{273883}{10946}a^{6}-\frac{131585}{10946}a^{5}-\frac{11556}{5473}a^{4}-\frac{10247}{10946}a^{3}-\frac{2003}{842}a^{2}-\frac{23489}{10946}a-\frac{15127}{10946}$ Copy content Toggle raw display (assuming GRH)
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:  \( 4608312.303502872 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 4608312.303502872 \cdot 1}{42\cdot\sqrt{10352754344108696148301025390625}}\cr\approx \mathstrut & 0.129099059619775 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - x^22 + 4*x^21 - 4*x^20 - 4*x^19 + 17*x^18 + 12*x^17 - 46*x^16 + 43*x^15 + 44*x^14 - 188*x^13 + 189*x^12 + 188*x^11 + 44*x^10 - 43*x^9 - 46*x^8 - 12*x^7 + 17*x^6 + 4*x^5 - 4*x^4 - 4*x^3 - x^2 + x + 1)
 
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^24 - x^23 - x^22 + 4*x^21 - 4*x^20 - 4*x^19 + 17*x^18 + 12*x^17 - 46*x^16 + 43*x^15 + 44*x^14 - 188*x^13 + 189*x^12 + 188*x^11 + 44*x^10 - 43*x^9 - 46*x^8 - 12*x^7 + 17*x^6 + 4*x^5 - 4*x^4 - 4*x^3 - x^2 + x + 1, 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^24 - x^23 - x^22 + 4*x^21 - 4*x^20 - 4*x^19 + 17*x^18 + 12*x^17 - 46*x^16 + 43*x^15 + 44*x^14 - 188*x^13 + 189*x^12 + 188*x^11 + 44*x^10 - 43*x^9 - 46*x^8 - 12*x^7 + 17*x^6 + 4*x^5 - 4*x^4 - 4*x^3 - x^2 + x + 1);
 
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^24 - x^23 - x^22 + 4*x^21 - 4*x^20 - 4*x^19 + 17*x^18 + 12*x^17 - 46*x^16 + 43*x^15 + 44*x^14 - 188*x^13 + 189*x^12 + 188*x^11 + 44*x^10 - 43*x^9 - 46*x^8 - 12*x^7 + 17*x^6 + 4*x^5 - 4*x^4 - 4*x^3 - x^2 + x + 1);
 
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

$C_2^2\times C_6$ (as 24T3):

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 24
The 24 conjugacy class representatives for $C_2^2\times C_6$
Character table for $C_2^2\times C_6$ is not computed

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), 6.0.2100875.1, 6.6.56723625.1, 6.0.64827.1, 6.6.300125.1, \(\Q(\zeta_{7})\), \(\Q(\zeta_{21})^+\), 6.0.8103375.1, 8.0.121550625.1, 12.0.3217569633140625.2, 12.0.4413675765625.1, 12.0.3217569633140625.1, 12.12.3217569633140625.1, 12.0.3217569633140625.3, 12.0.65664686390625.1, \(\Q(\zeta_{21})\)

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 ${\href{/padicField/2.6.0.1}{6} }^{4}$ R R R ${\href{/padicField/11.6.0.1}{6} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{12}$ ${\href{/padicField/17.6.0.1}{6} }^{4}$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ ${\href{/padicField/23.6.0.1}{6} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{12}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{12}$ ${\href{/padicField/43.2.0.1}{2} }^{12}$ ${\href{/padicField/47.6.0.1}{6} }^{4}$ ${\href{/padicField/53.6.0.1}{6} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 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$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(5\) Copy content Toggle raw display 5.12.6.1$x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(7\) 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$$[\ ]_{6}^{2}$
7.12.10.1$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$