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Properties

Label 24.0.444...169.1
Degree $24$
Signature $[0, 12]$
Discriminant $4.446\times 10^{34}$
Root discriminant \(27.78\)
Ramified primes $7,13$
Class number $7$ (GRH)
Class group [7] (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

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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 + 3*x^21 - x^20 - 5*x^19 + 7*x^18 + 3*x^17 - 17*x^16 + 11*x^15 + 23*x^14 - 45*x^13 - x^12 - 90*x^11 + 92*x^10 + 88*x^9 - 272*x^8 + 96*x^7 + 448*x^6 - 640*x^5 - 256*x^4 + 1536*x^3 - 1024*x^2 - 2048*x + 4096)
 
gp: K = bnfinit(y^24 - y^23 - y^22 + 3*y^21 - y^20 - 5*y^19 + 7*y^18 + 3*y^17 - 17*y^16 + 11*y^15 + 23*y^14 - 45*y^13 - y^12 - 90*y^11 + 92*y^10 + 88*y^9 - 272*y^8 + 96*y^7 + 448*y^6 - 640*y^5 - 256*y^4 + 1536*y^3 - 1024*y^2 - 2048*y + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - x^23 - x^22 + 3*x^21 - x^20 - 5*x^19 + 7*x^18 + 3*x^17 - 17*x^16 + 11*x^15 + 23*x^14 - 45*x^13 - x^12 - 90*x^11 + 92*x^10 + 88*x^9 - 272*x^8 + 96*x^7 + 448*x^6 - 640*x^5 - 256*x^4 + 1536*x^3 - 1024*x^2 - 2048*x + 4096);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - x^23 - x^22 + 3*x^21 - x^20 - 5*x^19 + 7*x^18 + 3*x^17 - 17*x^16 + 11*x^15 + 23*x^14 - 45*x^13 - x^12 - 90*x^11 + 92*x^10 + 88*x^9 - 272*x^8 + 96*x^7 + 448*x^6 - 640*x^5 - 256*x^4 + 1536*x^3 - 1024*x^2 - 2048*x + 4096)
 

\( x^{24} - x^{23} - x^{22} + 3 x^{21} - x^{20} - 5 x^{19} + 7 x^{18} + 3 x^{17} - 17 x^{16} + 11 x^{15} + \cdots + 4096 \) 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:   \(44455984353110737022824200630534169\) \(\medspace = 7^{12}\cdot 13^{22}\) 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:  \(27.78\)
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:  $7^{1/2}13^{11/12}\approx 27.775619852565686$
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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $24$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(91=7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{91}(64,·)$, $\chi_{91}(1,·)$, $\chi_{91}(69,·)$, $\chi_{91}(6,·)$, $\chi_{91}(71,·)$, $\chi_{91}(8,·)$, $\chi_{91}(76,·)$, $\chi_{91}(15,·)$, $\chi_{91}(83,·)$, $\chi_{91}(20,·)$, $\chi_{91}(85,·)$, $\chi_{91}(22,·)$, $\chi_{91}(90,·)$, $\chi_{91}(27,·)$, $\chi_{91}(29,·)$, $\chi_{91}(34,·)$, $\chi_{91}(36,·)$, $\chi_{91}(41,·)$, $\chi_{91}(43,·)$, $\chi_{91}(48,·)$, $\chi_{91}(50,·)$, $\chi_{91}(55,·)$, $\chi_{91}(57,·)$, $\chi_{91}(62,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{14}-\frac{1}{8}a^{13}+\frac{3}{8}a^{12}-\frac{1}{8}a^{11}+\frac{3}{8}a^{10}-\frac{1}{8}a^{9}+\frac{3}{8}a^{8}-\frac{1}{8}a^{7}+\frac{3}{8}a^{6}-\frac{1}{8}a^{5}+\frac{3}{8}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{16}-\frac{1}{16}a^{15}-\frac{1}{16}a^{14}+\frac{3}{16}a^{13}-\frac{1}{16}a^{12}-\frac{5}{16}a^{11}+\frac{7}{16}a^{10}+\frac{3}{16}a^{9}-\frac{1}{16}a^{8}-\frac{5}{16}a^{7}+\frac{7}{16}a^{6}+\frac{3}{16}a^{5}-\frac{1}{16}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{17}-\frac{1}{32}a^{16}-\frac{1}{32}a^{15}+\frac{3}{32}a^{14}-\frac{1}{32}a^{13}-\frac{5}{32}a^{12}+\frac{7}{32}a^{11}+\frac{3}{32}a^{10}+\frac{15}{32}a^{9}+\frac{11}{32}a^{8}-\frac{9}{32}a^{7}-\frac{13}{32}a^{6}-\frac{1}{32}a^{5}+\frac{3}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{18}-\frac{1}{64}a^{17}-\frac{1}{64}a^{16}+\frac{3}{64}a^{15}-\frac{1}{64}a^{14}-\frac{5}{64}a^{13}+\frac{7}{64}a^{12}+\frac{3}{64}a^{11}-\frac{17}{64}a^{10}+\frac{11}{64}a^{9}+\frac{23}{64}a^{8}+\frac{19}{64}a^{7}-\frac{1}{64}a^{6}-\frac{13}{32}a^{5}+\frac{7}{16}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{128}a^{19}-\frac{1}{128}a^{18}-\frac{1}{128}a^{17}+\frac{3}{128}a^{16}-\frac{1}{128}a^{15}-\frac{5}{128}a^{14}+\frac{7}{128}a^{13}+\frac{3}{128}a^{12}-\frac{17}{128}a^{11}+\frac{11}{128}a^{10}+\frac{23}{128}a^{9}-\frac{45}{128}a^{8}-\frac{1}{128}a^{7}+\frac{19}{64}a^{6}-\frac{9}{32}a^{5}-\frac{5}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{256}a^{20}-\frac{1}{256}a^{19}-\frac{1}{256}a^{18}+\frac{3}{256}a^{17}-\frac{1}{256}a^{16}-\frac{5}{256}a^{15}+\frac{7}{256}a^{14}+\frac{3}{256}a^{13}-\frac{17}{256}a^{12}+\frac{11}{256}a^{11}+\frac{23}{256}a^{10}-\frac{45}{256}a^{9}-\frac{1}{256}a^{8}-\frac{45}{128}a^{7}+\frac{23}{64}a^{6}+\frac{11}{32}a^{5}-\frac{1}{16}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{512}a^{21}-\frac{1}{512}a^{20}-\frac{1}{512}a^{19}+\frac{3}{512}a^{18}-\frac{1}{512}a^{17}-\frac{5}{512}a^{16}+\frac{7}{512}a^{15}+\frac{3}{512}a^{14}-\frac{17}{512}a^{13}+\frac{11}{512}a^{12}+\frac{23}{512}a^{11}-\frac{45}{512}a^{10}-\frac{1}{512}a^{9}-\frac{45}{256}a^{8}+\frac{23}{128}a^{7}+\frac{11}{64}a^{6}+\frac{15}{32}a^{5}+\frac{3}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{1024}a^{22}-\frac{1}{1024}a^{21}-\frac{1}{1024}a^{20}+\frac{3}{1024}a^{19}-\frac{1}{1024}a^{18}-\frac{5}{1024}a^{17}+\frac{7}{1024}a^{16}+\frac{3}{1024}a^{15}-\frac{17}{1024}a^{14}+\frac{11}{1024}a^{13}+\frac{23}{1024}a^{12}-\frac{45}{1024}a^{11}-\frac{1}{1024}a^{10}-\frac{45}{512}a^{9}+\frac{23}{256}a^{8}+\frac{11}{128}a^{7}-\frac{17}{64}a^{6}+\frac{3}{32}a^{5}+\frac{7}{16}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{2048}a^{23}-\frac{1}{2048}a^{22}-\frac{1}{2048}a^{21}+\frac{3}{2048}a^{20}-\frac{1}{2048}a^{19}-\frac{5}{2048}a^{18}+\frac{7}{2048}a^{17}+\frac{3}{2048}a^{16}-\frac{17}{2048}a^{15}+\frac{11}{2048}a^{14}+\frac{23}{2048}a^{13}-\frac{45}{2048}a^{12}-\frac{1}{2048}a^{11}-\frac{45}{1024}a^{10}+\frac{23}{512}a^{9}+\frac{11}{256}a^{8}-\frac{17}{128}a^{7}+\frac{3}{64}a^{6}+\frac{7}{32}a^{5}-\frac{5}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$ 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

$C_{7}$, which has order $7$ (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:   \( -\frac{1}{2} a^{14} + \frac{91}{2} a \)  (order $26$) 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:   $\frac{7}{128}a^{20}-\frac{627}{128}a^{7}+1$, $\frac{1}{2}a^{14}-\frac{91}{2}a+1$, $\frac{17}{512}a^{22}+\frac{1}{32}a^{18}-\frac{1541}{512}a^{9}-\frac{85}{32}a^{5}-1$, $\frac{7}{128}a^{20}+\frac{1}{2}a^{14}-\frac{627}{128}a^{7}-\frac{91}{2}a+1$, $\frac{11}{1024}a^{23}+\frac{7}{128}a^{20}-\frac{967}{1024}a^{10}-\frac{627}{128}a^{7}$, $\frac{45}{2048}a^{23}+\frac{45}{2048}a^{22}-\frac{135}{2048}a^{21}+\frac{45}{2048}a^{20}+\frac{161}{2048}a^{19}-\frac{315}{2048}a^{18}-\frac{135}{2048}a^{17}+\frac{509}{2048}a^{16}-\frac{495}{2048}a^{15}-\frac{2059}{2048}a^{14}+\frac{2025}{2048}a^{13}+\frac{45}{2048}a^{12}+\frac{1}{2048}a^{11}-\frac{1035}{512}a^{10}-\frac{495}{256}a^{9}+\frac{765}{128}a^{8}-\frac{135}{64}a^{7}-\frac{115}{16}a^{6}+\frac{225}{16}a^{5}+\frac{45}{8}a^{4}-\frac{177}{8}a^{3}+\frac{45}{2}a^{2}+\frac{181}{2}a-90$, $\frac{17}{512}a^{22}-\frac{1}{32}a^{19}-\frac{3}{8}a^{16}-\frac{1541}{512}a^{9}+\frac{85}{32}a^{6}+\frac{271}{8}a^{3}+1$, $\frac{1}{2048}a^{23}+\frac{23}{2048}a^{22}-\frac{69}{2048}a^{21}+\frac{23}{2048}a^{20}+\frac{339}{2048}a^{19}-\frac{161}{2048}a^{18}-\frac{69}{2048}a^{17}+\frac{391}{2048}a^{16}-\frac{1277}{2048}a^{15}-\frac{529}{2048}a^{14}+\frac{1035}{2048}a^{13}+\frac{23}{2048}a^{12}-\frac{45}{2048}a^{11}-\frac{91}{1024}a^{10}-\frac{253}{256}a^{9}+\frac{391}{128}a^{8}-\frac{69}{64}a^{7}-\frac{949}{64}a^{6}+\frac{115}{16}a^{5}+\frac{23}{8}a^{4}-\frac{69}{4}a^{3}+57a^{2}+23a-46$, $\frac{1}{32}a^{18}-\frac{1}{8}a^{17}-\frac{3}{8}a^{16}-\frac{1}{4}a^{15}-\frac{85}{32}a^{5}+\frac{93}{8}a^{4}+\frac{271}{8}a^{3}+\frac{89}{4}a^{2}$, $\frac{11}{1024}a^{23}+\frac{17}{256}a^{21}-\frac{1}{32}a^{19}-\frac{3}{16}a^{17}-\frac{967}{1024}a^{10}-\frac{1541}{256}a^{8}+\frac{85}{32}a^{6}+\frac{271}{16}a^{4}$, $\frac{11}{1024}a^{23}+\frac{3}{256}a^{22}-\frac{17}{512}a^{21}+\frac{5}{512}a^{20}+\frac{69}{512}a^{19}-\frac{135}{512}a^{18}-\frac{19}{512}a^{17}+\frac{97}{512}a^{16}-\frac{59}{512}a^{15}-\frac{135}{512}a^{14}+\frac{253}{512}a^{13}+\frac{17}{512}a^{12}-\frac{11}{512}a^{11}-\frac{1013}{1024}a^{10}-\frac{529}{512}a^{9}+\frac{771}{256}a^{8}-\frac{121}{128}a^{7}-\frac{391}{32}a^{6}+\frac{765}{32}a^{5}+\frac{51}{16}a^{4}-\frac{137}{8}a^{3}+\frac{43}{4}a^{2}+\frac{47}{2}a-45$ 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:  \( 42757649.54957395 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 42757649.54957395 \cdot 7}{26\cdot\sqrt{44455984353110737022824200630534169}}\cr\approx \mathstrut & 0.206695880097383 \end{aligned}\] (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 + 3*x^21 - x^20 - 5*x^19 + 7*x^18 + 3*x^17 - 17*x^16 + 11*x^15 + 23*x^14 - 45*x^13 - x^12 - 90*x^11 + 92*x^10 + 88*x^9 - 272*x^8 + 96*x^7 + 448*x^6 - 640*x^5 - 256*x^4 + 1536*x^3 - 1024*x^2 - 2048*x + 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 - x^23 - x^22 + 3*x^21 - x^20 - 5*x^19 + 7*x^18 + 3*x^17 - 17*x^16 + 11*x^15 + 23*x^14 - 45*x^13 - x^12 - 90*x^11 + 92*x^10 + 88*x^9 - 272*x^8 + 96*x^7 + 448*x^6 - 640*x^5 - 256*x^4 + 1536*x^3 - 1024*x^2 - 2048*x + 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 - x^23 - x^22 + 3*x^21 - x^20 - 5*x^19 + 7*x^18 + 3*x^17 - 17*x^16 + 11*x^15 + 23*x^14 - 45*x^13 - x^12 - 90*x^11 + 92*x^10 + 88*x^9 - 272*x^8 + 96*x^7 + 448*x^6 - 640*x^5 - 256*x^4 + 1536*x^3 - 1024*x^2 - 2048*x + 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 - x^23 - x^22 + 3*x^21 - x^20 - 5*x^19 + 7*x^18 + 3*x^17 - 17*x^16 + 11*x^15 + 23*x^14 - 45*x^13 - x^12 - 90*x^11 + 92*x^10 + 88*x^9 - 272*x^8 + 96*x^7 + 448*x^6 - 640*x^5 - 256*x^4 + 1536*x^3 - 1024*x^2 - 2048*x + 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

$C_2\times C_{12}$ (as 24T2):

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 $C_2\times C_{12}$
Character table for $C_2\times C_{12}$

Intermediate fields

\(\Q(\sqrt{-91}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-7}) \), 3.3.169.1, \(\Q(\sqrt{-7}, \sqrt{13})\), 4.4.107653.1, 4.0.2197.1, 6.0.127353499.1, \(\Q(\zeta_{13})^+\), 6.0.9796423.1, 8.0.11589168409.1, 12.0.16218913707543001.1, 12.12.210845878198059013.1, \(\Q(\zeta_{13})\)

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.12.0.1}{12} }^{2}$ ${\href{/padicField/3.6.0.1}{6} }^{4}$ ${\href{/padicField/5.4.0.1}{4} }^{6}$ R ${\href{/padicField/11.12.0.1}{12} }^{2}$ R ${\href{/padicField/17.6.0.1}{6} }^{4}$ ${\href{/padicField/19.12.0.1}{12} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{4}$ ${\href{/padicField/29.3.0.1}{3} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{6}$ ${\href{/padicField/37.12.0.1}{12} }^{2}$ ${\href{/padicField/41.12.0.1}{12} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{6}$ ${\href{/padicField/53.1.0.1}{1} }^{24}$ ${\href{/padicField/59.12.0.1}{12} }^{2}$

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 Deg $24$$2$$12$$12$
\(13\) Copy content Toggle raw display Deg $24$$12$$2$$22$