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Properties

Label 32.0.366...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $3.662\times 10^{47}$
Root discriminant \(30.65\)
Ramified primes $2,3,5,7$
Class number not computed
Class group not computed
Galois group $C_2^3\times C_4$ (as 32T34)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 3*x^30 + 4*x^28 - 9*x^26 + 27*x^24 - 93*x^22 + 188*x^20 - 279*x^18 + 581*x^16 - 1116*x^14 + 3008*x^12 - 5952*x^10 + 6912*x^8 - 9216*x^6 + 16384*x^4 - 49152*x^2 + 65536)
 
gp: K = bnfinit(y^32 - 3*y^30 + 4*y^28 - 9*y^26 + 27*y^24 - 93*y^22 + 188*y^20 - 279*y^18 + 581*y^16 - 1116*y^14 + 3008*y^12 - 5952*y^10 + 6912*y^8 - 9216*y^6 + 16384*y^4 - 49152*y^2 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 3*x^30 + 4*x^28 - 9*x^26 + 27*x^24 - 93*x^22 + 188*x^20 - 279*x^18 + 581*x^16 - 1116*x^14 + 3008*x^12 - 5952*x^10 + 6912*x^8 - 9216*x^6 + 16384*x^4 - 49152*x^2 + 65536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 3*x^30 + 4*x^28 - 9*x^26 + 27*x^24 - 93*x^22 + 188*x^20 - 279*x^18 + 581*x^16 - 1116*x^14 + 3008*x^12 - 5952*x^10 + 6912*x^8 - 9216*x^6 + 16384*x^4 - 49152*x^2 + 65536)
 

\( x^{32} - 3 x^{30} + 4 x^{28} - 9 x^{26} + 27 x^{24} - 93 x^{22} + 188 x^{20} - 279 x^{18} + 581 x^{16} + \cdots + 65536 \) Copy content Toggle raw display

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

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(366225584701948244050176000000000000000000000000\) \(\medspace = 2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\) 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:  \(30.65\)
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:  $2\cdot 3^{1/2}5^{3/4}7^{1/2}\approx 30.645530678223075$
Ramified primes:   \(2\), \(3\), \(5\), \(7\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $32$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(420=2^{2}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{420}(1,·)$, $\chi_{420}(391,·)$, $\chi_{420}(139,·)$, $\chi_{420}(13,·)$, $\chi_{420}(407,·)$, $\chi_{420}(281,·)$, $\chi_{420}(29,·)$, $\chi_{420}(419,·)$, $\chi_{420}(293,·)$, $\chi_{420}(167,·)$, $\chi_{420}(41,·)$, $\chi_{420}(43,·)$, $\chi_{420}(307,·)$, $\chi_{420}(181,·)$, $\chi_{420}(323,·)$, $\chi_{420}(197,·)$, $\chi_{420}(71,·)$, $\chi_{420}(209,·)$, $\chi_{420}(211,·)$, $\chi_{420}(349,·)$, $\chi_{420}(223,·)$, $\chi_{420}(97,·)$, $\chi_{420}(251,·)$, $\chi_{420}(337,·)$, $\chi_{420}(239,·)$, $\chi_{420}(113,·)$, $\chi_{420}(83,·)$, $\chi_{420}(169,·)$, $\chi_{420}(377,·)$, $\chi_{420}(379,·)$, $\chi_{420}(253,·)$, $\chi_{420}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{18}+\frac{1}{4}a^{16}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{19}+\frac{1}{8}a^{17}-\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{8}a^{9}+\frac{1}{8}a^{5}+\frac{1}{8}a^{3}$, $\frac{1}{176}a^{20}-\frac{1}{16}a^{18}+\frac{1}{4}a^{16}+\frac{5}{16}a^{14}-\frac{7}{16}a^{12}+\frac{43}{176}a^{10}-\frac{1}{4}a^{8}-\frac{5}{16}a^{6}+\frac{7}{16}a^{4}+\frac{1}{4}a^{2}-\frac{2}{11}$, $\frac{1}{352}a^{21}-\frac{1}{32}a^{19}+\frac{1}{8}a^{17}-\frac{11}{32}a^{15}+\frac{9}{32}a^{13}-\frac{133}{352}a^{11}-\frac{1}{8}a^{9}+\frac{11}{32}a^{7}-\frac{9}{32}a^{5}+\frac{1}{8}a^{3}+\frac{9}{22}a$, $\frac{1}{704}a^{22}+\frac{1}{704}a^{20}-\frac{1}{8}a^{18}+\frac{5}{64}a^{16}+\frac{5}{64}a^{14}+\frac{351}{704}a^{12}-\frac{29}{88}a^{10}+\frac{27}{64}a^{8}+\frac{27}{64}a^{6}-\frac{1}{8}a^{4}+\frac{5}{11}a^{2}+\frac{5}{11}$, $\frac{1}{1408}a^{23}+\frac{1}{1408}a^{21}-\frac{1}{16}a^{19}+\frac{5}{128}a^{17}-\frac{59}{128}a^{15}+\frac{351}{1408}a^{13}-\frac{29}{176}a^{11}+\frac{27}{128}a^{9}-\frac{37}{128}a^{7}+\frac{7}{16}a^{5}-\frac{3}{11}a^{3}+\frac{5}{22}a$, $\frac{1}{14080}a^{24}+\frac{1}{2816}a^{22}-\frac{1}{704}a^{20}-\frac{27}{1280}a^{18}-\frac{123}{256}a^{16}-\frac{1153}{2816}a^{14}-\frac{59}{3520}a^{12}+\frac{565}{2816}a^{10}+\frac{27}{256}a^{8}-\frac{7}{320}a^{6}-\frac{31}{88}a^{4}-\frac{3}{22}a^{2}-\frac{14}{55}$, $\frac{1}{28160}a^{25}+\frac{1}{5632}a^{23}-\frac{1}{1408}a^{21}-\frac{27}{2560}a^{19}-\frac{123}{512}a^{17}-\frac{1153}{5632}a^{15}-\frac{59}{7040}a^{13}+\frac{565}{5632}a^{11}+\frac{27}{512}a^{9}-\frac{7}{640}a^{7}+\frac{57}{176}a^{5}-\frac{3}{44}a^{3}-\frac{7}{55}a$, $\frac{1}{25062400}a^{26}+\frac{193}{25062400}a^{24}-\frac{237}{626560}a^{22}+\frac{27463}{25062400}a^{20}+\frac{48389}{2278400}a^{18}+\frac{1266179}{5012480}a^{16}+\frac{767593}{3132800}a^{14}-\frac{1794663}{25062400}a^{12}-\frac{1911467}{5012480}a^{10}+\frac{141669}{284800}a^{8}+\frac{411321}{1566400}a^{6}-\frac{20597}{78320}a^{4}-\frac{8869}{97900}a^{2}+\frac{11667}{24475}$, $\frac{1}{50124800}a^{27}+\frac{193}{50124800}a^{25}-\frac{237}{1253120}a^{23}+\frac{27463}{50124800}a^{21}+\frac{48389}{4556800}a^{19}+\frac{1266179}{10024960}a^{17}+\frac{767593}{6265600}a^{15}-\frac{1794663}{50124800}a^{13}-\frac{1911467}{10024960}a^{11}+\frac{141669}{569600}a^{9}+\frac{411321}{3132800}a^{7}-\frac{20597}{156640}a^{5}-\frac{8869}{195800}a^{3}+\frac{11667}{48950}a$, $\frac{1}{100249600}a^{28}+\frac{1}{100249600}a^{26}+\frac{413}{12531200}a^{24}-\frac{39177}{100249600}a^{22}+\frac{100983}{100249600}a^{20}-\frac{10419553}{100249600}a^{18}-\frac{13933}{140800}a^{16}+\frac{5837289}{100249600}a^{14}+\frac{12111721}{100249600}a^{12}+\frac{2466949}{12531200}a^{10}-\frac{628117}{1566400}a^{8}+\frac{741977}{1566400}a^{6}-\frac{143039}{391600}a^{4}+\frac{19079}{97900}a^{2}-\frac{651}{24475}$, $\frac{1}{200499200}a^{29}+\frac{1}{200499200}a^{27}+\frac{413}{25062400}a^{25}-\frac{39177}{200499200}a^{23}+\frac{100983}{200499200}a^{21}-\frac{10419553}{200499200}a^{19}-\frac{13933}{281600}a^{17}-\frac{94412311}{200499200}a^{15}-\frac{88137879}{200499200}a^{13}+\frac{2466949}{25062400}a^{11}-\frac{628117}{3132800}a^{9}-\frac{824423}{3132800}a^{7}+\frac{248561}{783200}a^{5}-\frac{78821}{195800}a^{3}-\frac{651}{48950}a$, $\frac{1}{400998400}a^{30}+\frac{1}{400998400}a^{28}-\frac{1}{50124800}a^{26}+\frac{5127}{400998400}a^{24}+\frac{170743}{400998400}a^{22}+\frac{581391}{400998400}a^{20}+\frac{3644777}{50124800}a^{18}-\frac{128966951}{400998400}a^{16}-\frac{162762007}{400998400}a^{14}+\frac{9092791}{50124800}a^{12}+\frac{9525477}{25062400}a^{10}-\frac{1900949}{6265600}a^{8}+\frac{655029}{1566400}a^{6}+\frac{55237}{195800}a^{4}-\frac{6217}{24475}a^{2}+\frac{11961}{24475}$, $\frac{1}{801996800}a^{31}+\frac{1}{801996800}a^{29}-\frac{1}{100249600}a^{27}+\frac{5127}{801996800}a^{25}+\frac{170743}{801996800}a^{23}+\frac{581391}{801996800}a^{21}+\frac{3644777}{100249600}a^{19}-\frac{128966951}{801996800}a^{17}-\frac{162762007}{801996800}a^{15}-\frac{41032009}{100249600}a^{13}+\frac{9525477}{50124800}a^{11}-\frac{1900949}{12531200}a^{9}-\frac{911371}{3132800}a^{7}-\frac{140563}{391600}a^{5}+\frac{9129}{24475}a^{3}+\frac{11961}{48950}a$ 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:  $2$

Class group and class number

not computed

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:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{1}{48950} a^{31} - \frac{7193}{48950} a \)  (order $60$) 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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 3*x^30 + 4*x^28 - 9*x^26 + 27*x^24 - 93*x^22 + 188*x^20 - 279*x^18 + 581*x^16 - 1116*x^14 + 3008*x^12 - 5952*x^10 + 6912*x^8 - 9216*x^6 + 16384*x^4 - 49152*x^2 + 65536)
 
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^32 - 3*x^30 + 4*x^28 - 9*x^26 + 27*x^24 - 93*x^22 + 188*x^20 - 279*x^18 + 581*x^16 - 1116*x^14 + 3008*x^12 - 5952*x^10 + 6912*x^8 - 9216*x^6 + 16384*x^4 - 49152*x^2 + 65536, 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^32 - 3*x^30 + 4*x^28 - 9*x^26 + 27*x^24 - 93*x^22 + 188*x^20 - 279*x^18 + 581*x^16 - 1116*x^14 + 3008*x^12 - 5952*x^10 + 6912*x^8 - 9216*x^6 + 16384*x^4 - 49152*x^2 + 65536);
 
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^32 - 3*x^30 + 4*x^28 - 9*x^26 + 27*x^24 - 93*x^22 + 188*x^20 - 279*x^18 + 581*x^16 - 1116*x^14 + 3008*x^12 - 5952*x^10 + 6912*x^8 - 9216*x^6 + 16384*x^4 - 49152*x^2 + 65536);
 
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

$C_2^3\times C_4$ (as 32T34):

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 32
The 32 conjugacy class representatives for $C_2^3\times C_4$
Character table for $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-15}) \), \(\Q(i, \sqrt{35})\), \(\Q(i, \sqrt{105})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{3}, \sqrt{35})\), \(\Q(\sqrt{-3}, \sqrt{35})\), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{3}, \sqrt{-35})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{7})\), \(\Q(i, \sqrt{21})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(\sqrt{-5}, \sqrt{-7})\), \(\Q(\sqrt{15}, \sqrt{21})\), \(\Q(\sqrt{-15}, \sqrt{-21})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-5}, \sqrt{7})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{15}, \sqrt{-21})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-5}, \sqrt{-21})\), \(\Q(\sqrt{7}, \sqrt{15})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{-21})\), \(\Q(\sqrt{-5}, \sqrt{21})\), \(\Q(\sqrt{7}, \sqrt{-15})\), \(\Q(\sqrt{-7}, \sqrt{15})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{7})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), 4.4.882000.1, 4.0.55125.1, \(\Q(\zeta_{15})^+\), 4.0.18000.1, \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{5})\), 4.4.6125.1, 4.0.98000.1, 8.0.31116960000.10, 8.0.384160000.1, 8.0.31116960000.4, 8.0.31116960000.8, 8.0.31116960000.7, 8.0.12960000.1, 8.0.49787136.1, 8.8.31116960000.1, 8.0.31116960000.1, 8.0.31116960000.9, 8.0.31116960000.3, 8.0.121550625.1, 8.0.31116960000.2, 8.0.31116960000.6, 8.0.31116960000.5, 8.0.777924000000.8, 8.0.324000000.1, \(\Q(\zeta_{20})\), 8.0.9604000000.1, 8.8.777924000000.3, 8.0.777924000000.3, 8.8.9604000000.1, 8.0.9604000000.3, 8.0.777924000000.4, 8.0.3038765625.2, 8.0.9604000000.2, 8.0.37515625.1, 8.8.777924000000.2, 8.0.3038765625.3, 8.8.3038765625.1, 8.0.777924000000.1, 8.0.777924000000.10, 8.0.777924000000.6, 8.0.777924000000.7, 8.0.777924000000.2, 8.8.777924000000.1, 8.0.777924000000.5, \(\Q(\zeta_{60})^+\), 8.0.324000000.3, 8.0.777924000000.9, 8.0.3038765625.1, \(\Q(\zeta_{15})\), 8.0.324000000.2, 16.0.968265199641600000000.1, 16.0.605165749776000000000000.4, 16.0.92236816000000000000.1, 16.0.605165749776000000000000.6, 16.0.605165749776000000000000.3, 16.0.605165749776000000000000.7, \(\Q(\zeta_{60})\), 16.16.605165749776000000000000.1, 16.0.605165749776000000000000.5, 16.0.605165749776000000000000.9, 16.0.605165749776000000000000.1, 16.0.605165749776000000000000.10, 16.0.9234096523681640625.1, 16.0.605165749776000000000000.8, 16.0.605165749776000000000000.2

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 R R R R ${\href{/padicField/11.2.0.1}{2} }^{16}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{16}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{16}$ ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{16}$

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 2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(7\) Copy content Toggle raw display 7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$