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Properties

Label 24.0.428...496.1
Degree $24$
Signature $[0, 12]$
Discriminant $4.287\times 10^{34}$
Root discriminant \(27.73\)
Ramified primes $2,3,23,107$
Class number $4$ (GRH)
Class group [4] (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 - 5*x^20 + 8*x^18 + 19*x^16 - 5*x^14 - 91*x^12 - 20*x^10 + 304*x^8 + 512*x^6 - 1280*x^4 - 2048*x^2 + 4096)
 
gp: K = bnfinit(y^24 - 2*y^22 - 5*y^20 + 8*y^18 + 19*y^16 - 5*y^14 - 91*y^12 - 20*y^10 + 304*y^8 + 512*y^6 - 1280*y^4 - 2048*y^2 + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 2*x^22 - 5*x^20 + 8*x^18 + 19*x^16 - 5*x^14 - 91*x^12 - 20*x^10 + 304*x^8 + 512*x^6 - 1280*x^4 - 2048*x^2 + 4096);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 2*x^22 - 5*x^20 + 8*x^18 + 19*x^16 - 5*x^14 - 91*x^12 - 20*x^10 + 304*x^8 + 512*x^6 - 1280*x^4 - 2048*x^2 + 4096)
 

\( x^{24} - 2 x^{22} - 5 x^{20} + 8 x^{18} + 19 x^{16} - 5 x^{14} - 91 x^{12} - 20 x^{10} + 304 x^{8} + \cdots + 4096 \) 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:   \(42870299553219194916193841757290496\) \(\medspace = 2^{24}\cdot 3^{12}\cdot 23^{4}\cdot 107^{8}\) 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:  \(27.73\)
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:  $2\cdot 3^{1/2}23^{1/2}107^{1/2}\approx 171.84877072589143$
Ramified primes:   \(2\), \(3\), \(23\), \(107\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{10}+\frac{1}{3}a^{6}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{6}a^{13}+\frac{1}{3}a^{11}-\frac{1}{2}a^{9}-\frac{1}{3}a^{7}-\frac{1}{2}a^{5}-\frac{1}{6}a^{3}+\frac{1}{6}a$, $\frac{1}{12}a^{14}-\frac{1}{6}a^{12}-\frac{5}{12}a^{10}+\frac{1}{3}a^{8}-\frac{1}{12}a^{6}-\frac{1}{12}a^{4}+\frac{5}{12}a^{2}-\frac{1}{3}$, $\frac{1}{24}a^{15}-\frac{1}{12}a^{13}-\frac{5}{24}a^{11}-\frac{1}{3}a^{9}+\frac{11}{24}a^{7}+\frac{11}{24}a^{5}+\frac{5}{24}a^{3}-\frac{1}{6}a$, $\frac{1}{48}a^{16}-\frac{1}{24}a^{14}-\frac{5}{48}a^{12}-\frac{1}{6}a^{10}-\frac{13}{48}a^{8}+\frac{11}{48}a^{6}+\frac{5}{48}a^{4}-\frac{1}{12}a^{2}$, $\frac{1}{96}a^{17}-\frac{1}{48}a^{15}-\frac{5}{96}a^{13}+\frac{5}{12}a^{11}-\frac{13}{96}a^{9}-\frac{37}{96}a^{7}+\frac{5}{96}a^{5}+\frac{11}{24}a^{3}-\frac{1}{2}a$, $\frac{1}{192}a^{18}-\frac{1}{96}a^{16}-\frac{5}{192}a^{14}+\frac{1}{24}a^{12}+\frac{19}{192}a^{10}+\frac{59}{192}a^{8}-\frac{9}{64}a^{6}+\frac{11}{48}a^{4}-\frac{1}{12}a^{2}+\frac{1}{3}$, $\frac{1}{384}a^{19}-\frac{1}{192}a^{17}-\frac{5}{384}a^{15}+\frac{1}{48}a^{13}+\frac{19}{384}a^{11}-\frac{133}{384}a^{9}+\frac{55}{128}a^{7}-\frac{37}{96}a^{5}+\frac{11}{24}a^{3}-\frac{1}{3}a$, $\frac{1}{108288}a^{20}+\frac{13}{54144}a^{18}-\frac{973}{108288}a^{16}+\frac{215}{27072}a^{14}+\frac{13891}{108288}a^{12}+\frac{16453}{36096}a^{10}-\frac{14231}{108288}a^{8}-\frac{959}{13536}a^{6}-\frac{649}{6768}a^{4}+\frac{809}{1692}a^{2}-\frac{137}{423}$, $\frac{1}{216576}a^{21}+\frac{13}{108288}a^{19}-\frac{973}{216576}a^{17}+\frac{215}{54144}a^{15}+\frac{13891}{216576}a^{13}-\frac{19643}{72192}a^{11}-\frac{14231}{216576}a^{9}-\frac{959}{27072}a^{7}+\frac{6119}{13536}a^{5}-\frac{883}{3384}a^{3}+\frac{143}{423}a$, $\frac{1}{9096192}a^{22}-\frac{1}{505344}a^{20}-\frac{11141}{9096192}a^{18}+\frac{8843}{1137024}a^{16}-\frac{83183}{3032064}a^{14}+\frac{114397}{1299456}a^{12}-\frac{352253}{1299456}a^{10}-\frac{195115}{758016}a^{8}-\frac{16775}{568512}a^{6}+\frac{41929}{142128}a^{4}+\frac{1829}{11844}a^{2}-\frac{4415}{8883}$, $\frac{1}{18192384}a^{23}-\frac{1}{1010688}a^{21}-\frac{11141}{18192384}a^{19}+\frac{8843}{2274048}a^{17}-\frac{83183}{6064128}a^{15}+\frac{114397}{2598912}a^{13}+\frac{947203}{2598912}a^{11}+\frac{562901}{1516032}a^{9}-\frac{16775}{1137024}a^{7}+\frac{41929}{284256}a^{5}-\frac{10015}{23688}a^{3}+\frac{2234}{8883}a$ 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

$C_{4}$, which has order $4$ (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:   \( \frac{5435}{4548096} a^{23} - \frac{15}{84224} a^{21} - \frac{28747}{4548096} a^{19} - \frac{3439}{1137024} a^{17} + \frac{24095}{1516032} a^{15} + \frac{12107}{649728} a^{13} - \frac{33055}{649728} a^{11} - \frac{54815}{379008} a^{9} + \frac{118229}{1137024} a^{7} + \frac{60995}{71064} a^{5} + \frac{227}{2961} a^{3} - \frac{21275}{8883} a \)  (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:   $\frac{4121}{2274048}a^{23}-\frac{529}{1516032}a^{21}-\frac{10685}{1137024}a^{19}-\frac{23045}{4548096}a^{17}+\frac{19513}{758016}a^{15}+\frac{22015}{649728}a^{13}-\frac{50981}{649728}a^{11}-\frac{261785}{1516032}a^{9}+\frac{173641}{1137024}a^{7}+\frac{44585}{35532}a^{5}+\frac{75}{2632}a^{3}-\frac{30613}{8883}a-1$, $\frac{22543}{4548096}a^{22}-\frac{961}{758016}a^{20}-\frac{128651}{4548096}a^{18}-\frac{785}{71064}a^{16}+\frac{36293}{505344}a^{14}+\frac{76795}{649728}a^{12}-\frac{161627}{649728}a^{10}-\frac{22707}{42112}a^{8}+\frac{3986}{8883}a^{6}+\frac{521795}{142128}a^{4}+\frac{215}{11844}a^{2}-\frac{89749}{8883}$, $\frac{739}{2274048}a^{22}-\frac{265}{189504}a^{20}+\frac{985}{2274048}a^{18}+\frac{2951}{1137024}a^{16}-\frac{2017}{758016}a^{14}-\frac{743}{324864}a^{12}-\frac{1289}{324864}a^{10}-\frac{8089}{379008}a^{8}+\frac{28897}{568512}a^{6}+\frac{23141}{71064}a^{4}-\frac{1439}{1974}a^{2}-\frac{2579}{8883}$, $\frac{14989}{2274048}a^{22}-\frac{185}{189504}a^{20}-\frac{82097}{2274048}a^{18}-\frac{18835}{1137024}a^{16}+\frac{8433}{84224}a^{14}+\frac{51295}{324864}a^{12}-\frac{103031}{324864}a^{10}-\frac{101069}{126336}a^{8}+\frac{319729}{568512}a^{6}+\frac{332699}{71064}a^{4}+\frac{4061}{5922}a^{2}-\frac{121217}{8883}$, $\frac{6023}{1010688}a^{22}+\frac{1921}{1516032}a^{20}-\frac{88145}{3032064}a^{18}-\frac{14405}{758016}a^{16}+\frac{229567}{3032064}a^{14}+\frac{69005}{433152}a^{12}-\frac{36023}{144384}a^{10}-\frac{259103}{379008}a^{8}+\frac{73123}{189504}a^{6}+\frac{95087}{23688}a^{4}+\frac{13165}{11844}a^{2}-\frac{31531}{2961}$, $\frac{317}{3032064}a^{22}-\frac{3413}{1516032}a^{20}+\frac{2725}{1010688}a^{18}+\frac{713}{126336}a^{16}-\frac{4793}{3032064}a^{14}-\frac{6263}{433152}a^{12}+\frac{1847}{433152}a^{10}+\frac{46211}{758016}a^{8}+\frac{2895}{21056}a^{6}-\frac{2357}{7896}a^{4}-\frac{3014}{2961}a^{2}+\frac{4813}{2961}$, $\frac{43775}{18192384}a^{23}-\frac{10847}{4548096}a^{22}-\frac{1853}{3032064}a^{21}+\frac{83}{379008}a^{20}-\frac{230155}{18192384}a^{19}+\frac{56359}{4548096}a^{18}-\frac{9575}{2274048}a^{17}+\frac{5645}{2274048}a^{16}+\frac{23719}{673792}a^{15}-\frac{55495}{1516032}a^{14}+\frac{146339}{2598912}a^{13}-\frac{29717}{649728}a^{12}-\frac{255571}{2598912}a^{11}+\frac{70393}{649728}a^{10}-\frac{133171}{505344}a^{9}+\frac{210515}{758016}a^{8}+\frac{125497}{568512}a^{7}-\frac{85157}{284256}a^{6}+\frac{225475}{142128}a^{5}-\frac{114005}{71064}a^{4}-\frac{1361}{11844}a^{3}+\frac{1}{7}a^{2}-\frac{78091}{17766}a+\frac{43349}{8883}$, $\frac{509}{1137024}a^{23}+\frac{1403}{4548096}a^{22}+\frac{349}{1516032}a^{21}-\frac{11}{189504}a^{20}-\frac{8321}{2274048}a^{19}-\frac{10267}{4548096}a^{18}-\frac{20155}{4548096}a^{17}-\frac{5135}{2274048}a^{16}+\frac{89}{8064}a^{15}-\frac{2879}{505344}a^{14}+\frac{21743}{649728}a^{13}-\frac{2395}{649728}a^{12}-\frac{191}{13824}a^{11}+\frac{15455}{649728}a^{10}-\frac{226567}{1516032}a^{9}+\frac{4755}{84224}a^{8}-\frac{84163}{1137024}a^{7}+\frac{4973}{71064}a^{6}+\frac{101735}{284256}a^{5}+\frac{6253}{35532}a^{4}+\frac{523}{1316}a^{3}-\frac{4307}{11844}a^{2}-\frac{9451}{17766}a-\frac{14828}{8883}$, $\frac{3203}{6064128}a^{23}-\frac{10847}{4548096}a^{22}+\frac{835}{1010688}a^{21}+\frac{83}{379008}a^{20}-\frac{7807}{6064128}a^{19}+\frac{56359}{4548096}a^{18}-\frac{173}{189504}a^{17}+\frac{5645}{2274048}a^{16}+\frac{221}{43008}a^{15}-\frac{55495}{1516032}a^{14}+\frac{4607}{866304}a^{13}-\frac{29717}{649728}a^{12}+\frac{127}{18432}a^{11}+\frac{70393}{649728}a^{10}-\frac{34715}{505344}a^{9}+\frac{210515}{758016}a^{8}-\frac{20893}{379008}a^{7}-\frac{85157}{284256}a^{6}+\frac{33869}{94752}a^{5}-\frac{114005}{71064}a^{4}+\frac{775}{2632}a^{3}+\frac{1}{7}a^{2}-\frac{13}{5922}a+\frac{34466}{8883}$, $\frac{4121}{2274048}a^{23}+\frac{647}{505344}a^{22}-\frac{529}{1516032}a^{21}-\frac{37}{84224}a^{20}-\frac{10685}{1137024}a^{19}-\frac{4835}{505344}a^{18}-\frac{23045}{4548096}a^{17}-\frac{295}{63168}a^{16}+\frac{19513}{758016}a^{15}+\frac{11749}{505344}a^{14}+\frac{22015}{649728}a^{13}+\frac{1835}{72192}a^{12}-\frac{50981}{649728}a^{11}-\frac{1305}{24064}a^{10}-\frac{261785}{1516032}a^{9}-\frac{7061}{42112}a^{8}+\frac{173641}{1137024}a^{7}+\frac{2741}{10528}a^{6}+\frac{44585}{35532}a^{5}+\frac{8635}{7896}a^{4}+\frac{75}{2632}a^{3}-\frac{755}{3948}a^{2}-\frac{30613}{8883}a-\frac{3068}{987}$, $\frac{1313}{2021376}a^{23}-\frac{1693}{3032064}a^{22}+\frac{2503}{3032064}a^{21}-\frac{4433}{1516032}a^{20}-\frac{14615}{6064128}a^{19}+\frac{3235}{1010688}a^{18}-\frac{5027}{1516032}a^{17}+\frac{737}{84224}a^{16}-\frac{3719}{6064128}a^{15}-\frac{2855}{3032064}a^{14}+\frac{229}{18432}a^{13}-\frac{10853}{433152}a^{12}+\frac{959}{288768}a^{11}-\frac{2323}{433152}a^{10}-\frac{1357}{16128}a^{9}+\frac{37003}{379008}a^{8}-\frac{3865}{47376}a^{7}+\frac{2255}{10528}a^{6}+\frac{29983}{94752}a^{5}-\frac{6179}{15792}a^{4}+\frac{1621}{5922}a^{3}-\frac{17921}{11844}a^{2}+\frac{82}{2961}a+\frac{2158}{2961}$ 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:  \( 52682650.63366059 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 52682650.63366059 \cdot 4}{12\cdot\sqrt{42870299553219194916193841757290496}}\cr\approx \mathstrut & 0.321089810701599 \end{aligned}\] (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 - 5*x^20 + 8*x^18 + 19*x^16 - 5*x^14 - 91*x^12 - 20*x^10 + 304*x^8 + 512*x^6 - 1280*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 - 5*x^20 + 8*x^18 + 19*x^16 - 5*x^14 - 91*x^12 - 20*x^10 + 304*x^8 + 512*x^6 - 1280*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 - 5*x^20 + 8*x^18 + 19*x^16 - 5*x^14 - 91*x^12 - 20*x^10 + 304*x^8 + 512*x^6 - 1280*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 - 5*x^20 + 8*x^18 + 19*x^16 - 5*x^14 - 91*x^12 - 20*x^10 + 304*x^8 + 512*x^6 - 1280*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 $C_2^3\times S_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), 3.3.321.1, \(\Q(\zeta_{12})\), 6.6.151676352.1, 6.0.6594624.1, 6.6.19783872.1, 6.0.455029056.1, 6.0.2369943.1, 6.6.7109829.1, 6.0.309123.1, 12.0.50549668409241.2, 12.0.23005715756027904.1, 12.0.207051441804251136.2, 12.12.207051441804251136.1, 12.0.207051441804251136.1, 12.0.391401591312384.1, 12.0.207051441804251136.3

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 ${\href{/padicField/5.6.0.1}{6} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.3.0.1}{3} }^{8}$ ${\href{/padicField/17.6.0.1}{6} }^{4}$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ R ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{12}$ ${\href{/padicField/47.2.0.1}{2} }^{12}$ ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{12}$

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}$
\(23\) Copy content Toggle raw display 23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(107\) 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}$