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Properties

Label 24.0.639...616.1
Degree $24$
Signature $[0, 12]$
Discriminant $6.399\times 10^{31}$
Root discriminant \(21.15\)
Ramified primes $2,3,71$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_2^3\times A_4$ (as 24T135)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 12*x^22 + 96*x^20 + 412*x^18 + 1260*x^16 + 2511*x^14 + 3666*x^12 + 3492*x^10 + 2322*x^8 + 736*x^6 + 165*x^4 + 15*x^2 + 1)
 
gp: K = bnfinit(y^24 + 12*y^22 + 96*y^20 + 412*y^18 + 1260*y^16 + 2511*y^14 + 3666*y^12 + 3492*y^10 + 2322*y^8 + 736*y^6 + 165*y^4 + 15*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 + 12*x^22 + 96*x^20 + 412*x^18 + 1260*x^16 + 2511*x^14 + 3666*x^12 + 3492*x^10 + 2322*x^8 + 736*x^6 + 165*x^4 + 15*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 + 12*x^22 + 96*x^20 + 412*x^18 + 1260*x^16 + 2511*x^14 + 3666*x^12 + 3492*x^10 + 2322*x^8 + 736*x^6 + 165*x^4 + 15*x^2 + 1)
 

\( x^{24} + 12 x^{22} + 96 x^{20} + 412 x^{18} + 1260 x^{16} + 2511 x^{14} + 3666 x^{12} + 3492 x^{10} + \cdots + 1 \) 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:   \(63990935712336613969356040175616\) \(\medspace = 2^{24}\cdot 3^{36}\cdot 71^{4}\) 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:  \(21.15\)
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:  $2\cdot 3^{3/2}71^{1/2}\approx 87.56711711595854$
Ramified primes:   \(2\), \(3\), \(71\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{9}a^{12}-\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{9}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{9}$, $\frac{1}{9}a^{13}-\frac{1}{3}a^{11}+\frac{1}{3}a^{9}-\frac{1}{9}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{9}a$, $\frac{1}{9}a^{14}+\frac{1}{3}a^{10}-\frac{1}{9}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{9}a^{2}+\frac{1}{3}$, $\frac{1}{9}a^{15}+\frac{1}{3}a^{11}-\frac{1}{9}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{9}a^{3}+\frac{1}{3}a$, $\frac{1}{9}a^{16}-\frac{1}{9}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{1}{9}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{9}a^{17}-\frac{1}{9}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{9}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{18}+\frac{1}{9}$, $\frac{1}{9}a^{19}+\frac{1}{9}a$, $\frac{1}{9}a^{20}+\frac{1}{9}a^{2}$, $\frac{1}{9}a^{21}+\frac{1}{9}a^{3}$, $\frac{1}{22509084141}a^{22}+\frac{299240297}{22509084141}a^{20}+\frac{1086357436}{22509084141}a^{18}+\frac{263275069}{7503028047}a^{16}+\frac{233460403}{7503028047}a^{14}+\frac{8471728}{2501009349}a^{12}+\frac{1770725249}{7503028047}a^{10}+\frac{729926189}{7503028047}a^{8}+\frac{400644671}{833669783}a^{6}+\frac{2775172954}{22509084141}a^{4}-\frac{6956139283}{22509084141}a^{2}+\frac{8821836493}{22509084141}$, $\frac{1}{22509084141}a^{23}+\frac{299240297}{22509084141}a^{21}+\frac{1086357436}{22509084141}a^{19}+\frac{263275069}{7503028047}a^{17}+\frac{233460403}{7503028047}a^{15}+\frac{8471728}{2501009349}a^{13}+\frac{1770725249}{7503028047}a^{11}+\frac{729926189}{7503028047}a^{9}+\frac{400644671}{833669783}a^{7}+\frac{2775172954}{22509084141}a^{5}-\frac{6956139283}{22509084141}a^{3}+\frac{8821836493}{22509084141}a$ 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

$C_{2}$, which has order $2$ (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:   \( \frac{11444877047}{22509084141} a^{23} + \frac{137633056654}{22509084141} a^{21} + \frac{1101716954711}{22509084141} a^{19} + \frac{526405153627}{2501009349} a^{17} + \frac{1610660658046}{2501009349} a^{15} + \frac{3214193264483}{2501009349} a^{13} + \frac{4686196269086}{2501009349} a^{11} + \frac{4458303483595}{2501009349} a^{9} + \frac{976928675150}{833669783} a^{7} + \frac{8125534199504}{22509084141} a^{5} + \frac{1531536208654}{22509084141} a^{3} + \frac{120679604099}{22509084141} a \)  (order $36$) 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:   $\frac{619953049}{7503028047}a^{22}+\frac{7514176526}{7503028047}a^{20}+\frac{20173630681}{2501009349}a^{18}+\frac{263883882577}{7503028047}a^{16}+\frac{273976317935}{2501009349}a^{14}+\frac{563862953302}{2501009349}a^{12}+\frac{2571714078989}{7503028047}a^{10}+\frac{875272055747}{2501009349}a^{8}+\frac{631036471396}{2501009349}a^{6}+\frac{736963751354}{7503028047}a^{4}+\frac{155975900984}{7503028047}a^{2}+\frac{6327972157}{2501009349}$, $\frac{67114523}{7503028047}a^{22}+\frac{149914040}{2501009349}a^{20}+\frac{2038768285}{7503028047}a^{18}-\frac{7739671063}{7503028047}a^{16}-\frac{70525495736}{7503028047}a^{14}-\frac{300610603153}{7503028047}a^{12}-\frac{677076731279}{7503028047}a^{10}-\frac{1020103358965}{7503028047}a^{8}-\frac{920759164553}{7503028047}a^{6}-\frac{508253912375}{7503028047}a^{4}-\frac{76950919601}{7503028047}a^{2}-\frac{789881359}{833669783}$, $\frac{179328331}{1184688639}a^{22}+\frac{2113972346}{1184688639}a^{20}+\frac{16774527445}{1184688639}a^{18}+\frac{23476172569}{394896213}a^{16}+\frac{70611182954}{394896213}a^{14}+\frac{136419366641}{394896213}a^{12}+\frac{193922607512}{394896213}a^{10}+\frac{174774979633}{394896213}a^{8}+\frac{108915911011}{394896213}a^{6}+\frac{76478728432}{1184688639}a^{4}+\frac{11436822773}{1184688639}a^{2}+\frac{53258611}{1184688639}$, $\frac{2028254029}{22509084141}a^{23}+\frac{619953049}{7503028047}a^{22}+\frac{25603108310}{22509084141}a^{21}+\frac{7514176526}{7503028047}a^{20}+\frac{209702589451}{22509084141}a^{19}+\frac{20173630681}{2501009349}a^{18}+\frac{318345715076}{7503028047}a^{17}+\frac{263883882577}{7503028047}a^{16}+\frac{1020497983090}{7503028047}a^{15}+\frac{273976317935}{2501009349}a^{14}+\frac{2209819920044}{7503028047}a^{13}+\frac{563862953302}{2501009349}a^{12}+\frac{3485022361084}{7503028047}a^{11}+\frac{2571714078989}{7503028047}a^{10}+\frac{3819801447053}{7503028047}a^{9}+\frac{875272055747}{2501009349}a^{8}+\frac{2927879855662}{7503028047}a^{7}+\frac{631036471396}{2501009349}a^{6}+\frac{4114846807807}{22509084141}a^{5}+\frac{736963751354}{7503028047}a^{4}+\frac{1007832627275}{22509084141}a^{3}+\frac{155975900984}{7503028047}a^{2}+\frac{110574306400}{22509084141}a+\frac{3826962808}{2501009349}$, $\frac{11444877047}{22509084141}a^{23}+\frac{137633056654}{22509084141}a^{21}+\frac{1101716954711}{22509084141}a^{19}+\frac{526405153627}{2501009349}a^{17}+\frac{1610660658046}{2501009349}a^{15}+\frac{3214193264483}{2501009349}a^{13}+\frac{4686196269086}{2501009349}a^{11}+\frac{4458303483595}{2501009349}a^{9}+\frac{976928675150}{833669783}a^{7}+\frac{8125534199504}{22509084141}a^{5}+\frac{1531536208654}{22509084141}a^{3}+\frac{120679604099}{22509084141}a+1$, $\frac{9598911082}{22509084141}a^{23}+\frac{141192132}{833669783}a^{22}+\frac{114668742269}{22509084141}a^{21}+\frac{14907423664}{7503028047}a^{20}+\frac{915560802184}{22509084141}a^{19}+\frac{117841199329}{7503028047}a^{18}+\frac{1302602289268}{7503028047}a^{17}+\frac{163448105309}{2501009349}a^{16}+\frac{3967233672409}{7503028047}a^{15}+\frac{1457777615708}{7503028047}a^{14}+\frac{2612906825936}{2501009349}a^{13}+\frac{2757148133629}{7503028047}a^{12}+\frac{11351583772973}{7503028047}a^{11}+\frac{1268888382104}{2501009349}a^{10}+\frac{10607924463677}{7503028047}a^{9}+\frac{3249635685580}{7503028047}a^{8}+\frac{2303544955898}{2501009349}a^{7}+\frac{1888193023526}{7503028047}a^{6}+\frac{6004815917653}{22509084141}a^{5}+\frac{34905016028}{833669783}a^{4}+\frac{1364976531563}{22509084141}a^{3}+\frac{19310119493}{2501009349}a^{2}+\frac{82487374321}{22509084141}a-\frac{4249634596}{7503028047}$, $\frac{4400644084}{22509084141}a^{22}+\frac{52766640764}{22509084141}a^{20}+\frac{421202850778}{22509084141}a^{18}+\frac{200046232753}{2501009349}a^{16}+\frac{202260710096}{833669783}a^{14}+\frac{3579094459925}{7503028047}a^{12}+\frac{567441198584}{833669783}a^{10}+\frac{520782153364}{833669783}a^{8}+\frac{2936876666125}{7503028047}a^{6}+\frac{2368540968586}{22509084141}a^{4}+\frac{476158402898}{22509084141}a^{2}+\frac{20664914110}{22509084141}$, $\frac{197459503}{2501009349}a^{23}-\frac{621035899}{2501009349}a^{22}+\frac{7386137246}{7503028047}a^{21}-\frac{7431186970}{2501009349}a^{20}+\frac{20026156261}{2501009349}a^{19}-\frac{177919668112}{7503028047}a^{18}+\frac{269145601234}{7503028047}a^{17}-\frac{759489917161}{7503028047}a^{16}+\frac{848094153343}{7503028047}a^{15}-\frac{256207974339}{833669783}a^{14}+\frac{1779928507439}{7503028047}a^{13}-\frac{1512358880356}{2501009349}a^{12}+\frac{2695374398285}{7503028047}a^{11}-\frac{6502039544597}{7503028047}a^{10}+\frac{2741330290241}{7503028047}a^{9}-\frac{1996988658607}{2501009349}a^{8}+\frac{1892969686828}{7503028047}a^{7}-\frac{1260387570118}{2501009349}a^{6}+\frac{697668712216}{7503028047}a^{5}-\frac{1011813856468}{7503028047}a^{4}+\frac{13709276671}{833669783}a^{3}-\frac{60437949506}{2501009349}a^{2}+\frac{14017008989}{7503028047}a-\frac{9767374393}{7503028047}$, $\frac{661723195}{7503028047}a^{22}+\frac{7312056928}{7503028047}a^{20}+\frac{18668760433}{2501009349}a^{18}+\frac{212797115156}{7503028047}a^{16}+\frac{579708027520}{7503028047}a^{14}+\frac{899635430699}{7503028047}a^{12}+\frac{958657268002}{7503028047}a^{10}+\frac{289562026973}{7503028047}a^{8}-\frac{194665931867}{7503028047}a^{6}-\frac{156582919658}{2501009349}a^{4}-\frac{32235301585}{7503028047}a^{2}-\frac{3111585535}{7503028047}$, $\frac{7075936745}{22509084141}a^{23}-\frac{1439848867}{22509084141}a^{22}+\frac{85256669632}{22509084141}a^{21}-\frac{17239462259}{22509084141}a^{20}+\frac{682498159730}{22509084141}a^{19}-\frac{137649958558}{22509084141}a^{18}+\frac{326407962364}{2501009349}a^{17}-\frac{196037069224}{7503028047}a^{16}+\frac{2991412415681}{7503028047}a^{15}-\frac{595624532377}{7503028047}a^{14}+\frac{5955709962028}{7503028047}a^{13}-\frac{1171699259645}{7503028047}a^{12}+\frac{957684754748}{833669783}a^{11}-\frac{1672048863182}{7503028047}a^{10}+\frac{8104903973056}{7503028047}a^{9}-\frac{1511966581406}{7503028047}a^{8}+\frac{5193594123308}{7503028047}a^{7}-\frac{894176071597}{7503028047}a^{6}+\frac{4491750183740}{22509084141}a^{5}-\frac{472658779483}{22509084141}a^{4}+\frac{707498939971}{22509084141}a^{3}+\frac{66568436785}{22509084141}a^{2}+\frac{76446225407}{22509084141}a+\frac{11345858663}{22509084141}$, $\frac{3265838995}{22509084141}a^{23}-\frac{7089829927}{22509084141}a^{22}+\frac{37215781391}{22509084141}a^{21}-\frac{84834884825}{22509084141}a^{20}+\frac{290203642360}{22509084141}a^{19}-\frac{677904683332}{22509084141}a^{18}+\frac{128961296026}{2501009349}a^{17}-\frac{966494598056}{7503028047}a^{16}+\frac{1112883089510}{7503028047}a^{15}-\frac{327621451973}{833669783}a^{14}+\frac{1957966652141}{7503028047}a^{13}-\frac{5843439506293}{7503028047}a^{12}+\frac{831793803671}{2501009349}a^{11}-\frac{8483871837436}{7503028047}a^{10}+\frac{1685838057757}{7503028047}a^{9}-\frac{2654578051063}{2501009349}a^{8}+\frac{600424435342}{7503028047}a^{7}-\frac{5204980328840}{7503028047}a^{6}-\frac{1290428063573}{22509084141}a^{5}-\frac{4581923155951}{22509084141}a^{4}-\frac{524669890123}{22509084141}a^{3}-\frac{1008866965832}{22509084141}a^{2}-\frac{191347569218}{22509084141}a-\frac{72696660901}{22509084141}$ 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:  \( 7833899.703521124 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 7833899.703521124 \cdot 2}{36\cdot\sqrt{63990935712336613969356040175616}}\cr\approx \mathstrut & 0.205970342829836 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 + 12*x^22 + 96*x^20 + 412*x^18 + 1260*x^16 + 2511*x^14 + 3666*x^12 + 3492*x^10 + 2322*x^8 + 736*x^6 + 165*x^4 + 15*x^2 + 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 + 12*x^22 + 96*x^20 + 412*x^18 + 1260*x^16 + 2511*x^14 + 3666*x^12 + 3492*x^10 + 2322*x^8 + 736*x^6 + 165*x^4 + 15*x^2 + 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 + 12*x^22 + 96*x^20 + 412*x^18 + 1260*x^16 + 2511*x^14 + 3666*x^12 + 3492*x^10 + 2322*x^8 + 736*x^6 + 165*x^4 + 15*x^2 + 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 + 12*x^22 + 96*x^20 + 412*x^18 + 1260*x^16 + 2511*x^14 + 3666*x^12 + 3492*x^10 + 2322*x^8 + 736*x^6 + 165*x^4 + 15*x^2 + 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

$C_2^3\times A_4$ (as 24T135):

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 96
The 32 conjugacy class representatives for $C_2^3\times A_4$
Character table for $C_2^3\times A_4$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{12})\), 6.0.465831.1, \(\Q(\zeta_{36})^+\), \(\Q(\zeta_{9})\), 6.6.29813184.1, 6.0.89439552.2, 6.6.1397493.1, 6.0.419904.1, 12.0.7999433461960704.1, 12.0.7999433461960704.2, 12.0.7999433461960704.3, 12.0.1952986685049.1, 12.12.7999433461960704.1, 12.0.888825940217856.1, \(\Q(\zeta_{36})\)

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 24 siblings: data not computed
Degree 32 sibling: data not computed
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 R R ${\href{/padicField/5.6.0.1}{6} }^{4}$ ${\href{/padicField/7.6.0.1}{6} }^{4}$ ${\href{/padicField/11.6.0.1}{6} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{12}$ ${\href{/padicField/19.2.0.1}{2} }^{12}$ ${\href{/padicField/23.6.0.1}{6} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }^{8}$ ${\href{/padicField/41.6.0.1}{6} }^{4}$ ${\href{/padicField/43.6.0.1}{6} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{12}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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
\(2\) Copy content Toggle raw display 2.12.12.26$x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
\(3\) Copy content Toggle raw display 3.12.18.82$x^{12} + 24 x^{11} + 252 x^{10} + 1558 x^{9} + 6450 x^{8} + 19068 x^{7} + 41627 x^{6} + 68094 x^{5} + 83298 x^{4} + 74306 x^{3} + 45618 x^{2} + 17400 x + 3277$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
3.12.18.82$x^{12} + 24 x^{11} + 252 x^{10} + 1558 x^{9} + 6450 x^{8} + 19068 x^{7} + 41627 x^{6} + 68094 x^{5} + 83298 x^{4} + 74306 x^{3} + 45618 x^{2} + 17400 x + 3277$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
\(71\) Copy content Toggle raw display 71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$