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Properties

Label 24.0.141...576.1
Degree $24$
Signature $[0, 12]$
Discriminant $1.415\times 10^{35}$
Root discriminant \(29.15\)
Ramified primes $2,3,31,107$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $C_2^3\times S_4$ (as 24T400)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 2*x^22 + 3*x^20 + 12*x^18 + 7*x^16 + 11*x^14 + 85*x^12 + 44*x^10 + 112*x^8 + 768*x^6 + 768*x^4 + 2048*x^2 + 4096)
 
gp: K = bnfinit(y^24 + 2*y^22 + 3*y^20 + 12*y^18 + 7*y^16 + 11*y^14 + 85*y^12 + 44*y^10 + 112*y^8 + 768*y^6 + 768*y^4 + 2048*y^2 + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 + 2*x^22 + 3*x^20 + 12*x^18 + 7*x^16 + 11*x^14 + 85*x^12 + 44*x^10 + 112*x^8 + 768*x^6 + 768*x^4 + 2048*x^2 + 4096);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 + 2*x^22 + 3*x^20 + 12*x^18 + 7*x^16 + 11*x^14 + 85*x^12 + 44*x^10 + 112*x^8 + 768*x^6 + 768*x^4 + 2048*x^2 + 4096)
 

\( x^{24} + 2 x^{22} + 3 x^{20} + 12 x^{18} + 7 x^{16} + 11 x^{14} + 85 x^{12} + 44 x^{10} + 112 x^{8} + \cdots + 4096 \) 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:   \(141478989546523004521132546458648576\) \(\medspace = 2^{24}\cdot 3^{12}\cdot 31^{4}\cdot 107^{8}\) 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:  \(29.15\)
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:  $2\cdot 3^{1/2}31^{1/2}107^{1/2}\approx 199.50939827486823$
Ramified primes:   \(2\), \(3\), \(31\), \(107\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{10}+\frac{1}{3}a^{6}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{6}a^{13}+\frac{1}{3}a^{11}-\frac{1}{2}a^{9}-\frac{1}{3}a^{7}-\frac{1}{2}a^{5}-\frac{1}{6}a^{3}+\frac{1}{6}a$, $\frac{1}{36}a^{14}-\frac{1}{6}a^{12}-\frac{1}{36}a^{10}+\frac{4}{9}a^{8}-\frac{17}{36}a^{6}-\frac{13}{36}a^{4}+\frac{1}{4}a^{2}+\frac{1}{9}$, $\frac{1}{72}a^{15}-\frac{1}{12}a^{13}+\frac{35}{72}a^{11}-\frac{5}{18}a^{9}-\frac{17}{72}a^{7}-\frac{13}{72}a^{5}-\frac{3}{8}a^{3}+\frac{1}{18}a$, $\frac{1}{144}a^{16}-\frac{1}{72}a^{14}+\frac{11}{144}a^{12}+\frac{1}{3}a^{10}-\frac{25}{144}a^{8}+\frac{7}{16}a^{6}-\frac{7}{144}a^{4}+\frac{5}{18}a^{2}+\frac{1}{9}$, $\frac{1}{288}a^{17}-\frac{1}{144}a^{15}+\frac{11}{288}a^{13}-\frac{1}{3}a^{11}-\frac{25}{288}a^{9}-\frac{9}{32}a^{7}-\frac{7}{288}a^{5}+\frac{5}{36}a^{3}+\frac{1}{18}a$, $\frac{1}{1728}a^{18}+\frac{1}{864}a^{16}+\frac{1}{576}a^{14}-\frac{13}{432}a^{12}-\frac{121}{1728}a^{10}-\frac{181}{1728}a^{8}-\frac{43}{1728}a^{6}-\frac{71}{144}a^{4}-\frac{25}{108}a^{2}+\frac{1}{27}$, $\frac{1}{3456}a^{19}+\frac{1}{1728}a^{17}+\frac{1}{1152}a^{15}-\frac{13}{864}a^{13}-\frac{121}{3456}a^{11}-\frac{181}{3456}a^{9}-\frac{43}{3456}a^{7}+\frac{73}{288}a^{5}+\frac{83}{216}a^{3}+\frac{1}{54}a$, $\frac{1}{20736}a^{20}-\frac{1}{3456}a^{18}+\frac{35}{20736}a^{16}-\frac{43}{5184}a^{14}+\frac{823}{20736}a^{12}-\frac{3821}{20736}a^{10}-\frac{3251}{20736}a^{8}+\frac{1061}{5184}a^{6}-\frac{10}{81}a^{4}-\frac{1}{4}a^{2}+\frac{1}{81}$, $\frac{1}{41472}a^{21}-\frac{1}{6912}a^{19}+\frac{35}{41472}a^{17}-\frac{43}{10368}a^{15}+\frac{823}{41472}a^{13}-\frac{3821}{41472}a^{11}+\frac{17485}{41472}a^{9}+\frac{1061}{10368}a^{7}+\frac{71}{162}a^{5}-\frac{1}{8}a^{3}+\frac{1}{162}a$, $\frac{1}{248832}a^{22}-\frac{1}{124416}a^{20}+\frac{11}{248832}a^{18}-\frac{1}{7776}a^{16}+\frac{5}{9216}a^{14}-\frac{529}{248832}a^{12}+\frac{2201}{248832}a^{10}-\frac{365}{10368}a^{8}+\frac{2197}{15552}a^{6}+\frac{1703}{3888}a^{4}+\frac{61}{243}a^{2}+\frac{1}{243}$, $\frac{1}{497664}a^{23}-\frac{1}{248832}a^{21}+\frac{11}{497664}a^{19}-\frac{1}{15552}a^{17}+\frac{5}{18432}a^{15}-\frac{529}{497664}a^{13}+\frac{2201}{497664}a^{11}-\frac{365}{20736}a^{9}+\frac{2197}{31104}a^{7}-\frac{2185}{7776}a^{5}+\frac{61}{486}a^{3}-\frac{121}{243}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$

Class group and class number

$C_{3}$, which has order $3$ (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:   \( \frac{163}{248832} a^{23} - \frac{1}{31104} a^{21} + \frac{893}{248832} a^{19} + \frac{509}{124416} a^{17} - \frac{673}{82944} a^{15} + \frac{5567}{248832} a^{13} + \frac{3221}{248832} a^{11} - \frac{593}{13824} a^{9} + \frac{7727}{31104} a^{7} + \frac{283}{1944} a^{5} - \frac{205}{1944} a^{3} + \frac{535}{243} a \)  (order $12$) 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:   $\frac{7}{62208}a^{23}-\frac{125}{124416}a^{22}+\frac{11}{124416}a^{21}+\frac{37}{31104}a^{20}-\frac{65}{31104}a^{19}-\frac{43}{124416}a^{18}+\frac{125}{124416}a^{17}-\frac{361}{62208}a^{16}+\frac{115}{20736}a^{15}+\frac{367}{41472}a^{14}-\frac{1805}{124416}a^{13}-\frac{829}{124416}a^{12}+\frac{2071}{124416}a^{11}-\frac{7063}{124416}a^{10}+\frac{437}{41472}a^{9}+\frac{1055}{20736}a^{8}-\frac{1229}{15552}a^{7}-\frac{1243}{15552}a^{6}+\frac{689}{3888}a^{5}-\frac{1817}{3888}a^{4}+\frac{425}{1944}a^{3}+\frac{497}{972}a^{2}-\frac{191}{243}a-\frac{22}{243}$, $\frac{23}{82944}a^{22}+\frac{19}{41472}a^{20}+\frac{229}{82944}a^{18}+\frac{71}{20736}a^{16}+\frac{155}{27648}a^{14}+\frac{325}{82944}a^{12}-\frac{1253}{82944}a^{10}-\frac{83}{6912}a^{8}-\frac{5}{2592}a^{6}+\frac{143}{648}a^{4}+\frac{173}{324}a^{2}+\frac{74}{81}$, $\frac{335}{248832}a^{22}+\frac{379}{124416}a^{20}-\frac{275}{248832}a^{18}+\frac{455}{62208}a^{16}+\frac{1171}{82944}a^{14}-\frac{3635}{248832}a^{12}+\frac{17971}{248832}a^{10}+\frac{743}{6912}a^{8}-\frac{401}{1944}a^{6}+\frac{127}{243}a^{4}+\frac{1559}{972}a^{2}-\frac{19}{243}$, $\frac{1}{82944}a^{22}-\frac{7}{4608}a^{20}-\frac{13}{82944}a^{18}-\frac{37}{20736}a^{16}-\frac{1577}{82944}a^{14}+\frac{715}{82944}a^{12}-\frac{2459}{82944}a^{10}-\frac{1729}{20736}a^{8}+\frac{193}{2592}a^{6}-\frac{5}{432}a^{4}-\frac{113}{324}a^{2}+\frac{1}{27}$, $\frac{305}{124416}a^{22}+\frac{73}{62208}a^{20}+\frac{979}{124416}a^{18}+\frac{287}{31104}a^{16}-\frac{43}{4608}a^{14}+\frac{5515}{124416}a^{12}+\frac{9589}{124416}a^{10}-\frac{467}{10368}a^{8}+\frac{3779}{7776}a^{6}+\frac{2435}{3888}a^{4}+\frac{143}{486}a^{2}+\frac{1204}{243}$, $\frac{101}{82944}a^{23}+\frac{7}{5184}a^{21}+\frac{331}{82944}a^{19}+\frac{77}{13824}a^{17}+\frac{379}{82944}a^{15}+\frac{523}{27648}a^{13}+\frac{451}{9216}a^{11}+\frac{1525}{41472}a^{9}+\frac{467}{3456}a^{7}+\frac{1555}{2592}a^{5}+\frac{451}{648}a^{3}+\frac{317}{162}a-1$, $\frac{263}{248832}a^{23}+\frac{67}{124416}a^{22}+\frac{17}{62208}a^{21}-\frac{19}{62208}a^{20}+\frac{769}{248832}a^{19}+\frac{17}{124416}a^{18}+\frac{1147}{124416}a^{17}+\frac{1}{1944}a^{16}-\frac{23}{3072}a^{15}+\frac{247}{41472}a^{14}+\frac{7879}{248832}a^{13}+\frac{77}{124416}a^{12}+\frac{8101}{248832}a^{11}-\frac{1477}{124416}a^{10}-\frac{1765}{41472}a^{9}+\frac{239}{5184}a^{8}+\frac{7051}{31104}a^{7}-\frac{31}{243}a^{6}+\frac{1673}{7776}a^{5}+\frac{121}{3888}a^{4}+\frac{319}{1944}a^{3}+\frac{157}{486}a^{2}+\frac{515}{243}a-\frac{295}{243}$, $\frac{169}{248832}a^{23}+\frac{47}{27648}a^{22}-\frac{19}{31104}a^{21}+\frac{31}{13824}a^{20}-\frac{121}{248832}a^{19}+\frac{221}{27648}a^{18}+\frac{551}{124416}a^{17}+\frac{61}{6912}a^{16}-\frac{467}{82944}a^{15}-\frac{5}{1024}a^{14}+\frac{3197}{248832}a^{13}+\frac{503}{9216}a^{12}+\frac{815}{248832}a^{11}+\frac{697}{9216}a^{10}-\frac{1489}{41472}a^{9}-\frac{151}{6912}a^{8}+\frac{695}{31104}a^{7}+\frac{407}{864}a^{6}+\frac{145}{486}a^{5}+\frac{175}{432}a^{4}-\frac{229}{1944}a^{3}+\frac{47}{54}a^{2}-\frac{41}{243}a+\frac{16}{3}$, $\frac{569}{497664}a^{23}-\frac{103}{41472}a^{22}+\frac{235}{248832}a^{21}+\frac{1}{2592}a^{20}+\frac{2515}{497664}a^{19}-\frac{185}{41472}a^{18}+\frac{619}{62208}a^{17}-\frac{95}{6912}a^{16}-\frac{779}{165888}a^{15}+\frac{343}{41472}a^{14}+\frac{13807}{497664}a^{13}-\frac{65}{1536}a^{12}+\frac{27097}{497664}a^{11}-\frac{1699}{13824}a^{10}-\frac{37}{1728}a^{9}+\frac{2785}{20736}a^{8}+\frac{2357}{7776}a^{7}-\frac{599}{1728}a^{6}+\frac{253}{486}a^{5}-\frac{1223}{1296}a^{4}+\frac{131}{486}a^{3}+\frac{115}{324}a^{2}+\frac{778}{243}a-\frac{343}{81}$, $\frac{73}{41472}a^{22}+\frac{1}{288}a^{20}+\frac{95}{41472}a^{18}+\frac{187}{20736}a^{16}+\frac{511}{41472}a^{14}+\frac{373}{41472}a^{12}+\frac{1375}{41472}a^{10}+\frac{3805}{20736}a^{8}+\frac{101}{648}a^{6}+\frac{31}{54}a^{4}+\frac{155}{81}a^{2}+\frac{53}{27}$, $\frac{155}{55296}a^{23}-\frac{133}{248832}a^{22}-\frac{23}{82944}a^{21}+\frac{277}{124416}a^{20}+\frac{497}{55296}a^{19}-\frac{1175}{248832}a^{18}+\frac{823}{41472}a^{17}-\frac{5}{1944}a^{16}-\frac{3593}{165888}a^{15}+\frac{469}{27648}a^{14}+\frac{9419}{165888}a^{13}-\frac{6539}{248832}a^{12}+\frac{20597}{165888}a^{11}-\frac{5885}{248832}a^{10}-\frac{4777}{41472}a^{9}+\frac{1457}{10368}a^{8}+\frac{6547}{10368}a^{7}-\frac{4039}{15552}a^{6}+\frac{1469}{1296}a^{5}-\frac{61}{1944}a^{4}-\frac{25}{72}a^{3}+\frac{158}{243}a^{2}+\frac{445}{81}a-\frac{448}{243}$ 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:  \( 127557726.3707635 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 127557726.3707635 \cdot 3}{12\cdot\sqrt{141478989546523004521132546458648576}}\cr\approx \mathstrut & 0.320966143362790 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 + 2*x^22 + 3*x^20 + 12*x^18 + 7*x^16 + 11*x^14 + 85*x^12 + 44*x^10 + 112*x^8 + 768*x^6 + 768*x^4 + 2048*x^2 + 4096)
 
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 + 2*x^22 + 3*x^20 + 12*x^18 + 7*x^16 + 11*x^14 + 85*x^12 + 44*x^10 + 112*x^8 + 768*x^6 + 768*x^4 + 2048*x^2 + 4096, 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 + 2*x^22 + 3*x^20 + 12*x^18 + 7*x^16 + 11*x^14 + 85*x^12 + 44*x^10 + 112*x^8 + 768*x^6 + 768*x^4 + 2048*x^2 + 4096);
 
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 + 2*x^22 + 3*x^20 + 12*x^18 + 7*x^16 + 11*x^14 + 85*x^12 + 44*x^10 + 112*x^8 + 768*x^6 + 768*x^4 + 2048*x^2 + 4096);
 
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

$C_2^3\times S_4$ (as 24T400):

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 192
The 40 conjugacy class representatives for $C_2^3\times S_4$
Character table for $C_2^3\times S_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), 3.3.321.1, \(\Q(\zeta_{12})\), 6.0.6594624.1, 6.6.9582813.1, 6.0.3194271.1, 6.6.19783872.1, 6.0.613300032.1, 6.6.204433344.1, 6.0.309123.1, 12.0.376136929251201024.3, 12.0.376136929251201024.1, 12.0.376136929251201024.2, 12.0.41792992139022336.1, 12.12.376136929251201024.1, 12.0.391401591312384.1, 12.0.91830304992969.1

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 24 siblings: data not computed
Degree 32 siblings: 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 ${\href{/padicField/5.6.0.1}{6} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }^{4}$ ${\href{/padicField/17.6.0.1}{6} }^{4}$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{12}$ R ${\href{/padicField/37.6.0.1}{6} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{12}$ ${\href{/padicField/43.2.0.1}{2} }^{12}$ ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{12}$

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 3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(31\) Copy content Toggle raw display 31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.0.1$x^{4} + 3 x^{2} + 16 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.0.1$x^{4} + 3 x^{2} + 16 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.0.1$x^{4} + 3 x^{2} + 16 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.0.1$x^{4} + 3 x^{2} + 16 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
\(107\) Copy content Toggle raw display 107.2.0.1$x^{2} + 103 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} + 103 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} + 103 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} + 103 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
107.4.2.1$x^{4} + 206 x^{3} + 10827 x^{2} + 22454 x + 1146188$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 206 x^{3} + 10827 x^{2} + 22454 x + 1146188$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 206 x^{3} + 10827 x^{2} + 22454 x + 1146188$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 206 x^{3} + 10827 x^{2} + 22454 x + 1146188$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$