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Properties

Label 16.0.144...496.6
Degree $16$
Signature $[0, 8]$
Discriminant $1.446\times 10^{29}$
Root discriminant \(66.45\)
Ramified primes $2,3,23$
Class number $19200$ (GRH)
Class group [4, 40, 120] (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1366*x^12 - 4920*x^11 + 18684*x^10 - 54040*x^9 + 169095*x^8 - 396536*x^7 + 1046676*x^6 - 1935192*x^5 + 4326842*x^4 - 5804824*x^3 + 11026220*x^2 - 8393144*x + 13286974)
 
gp: K = bnfinit(y^16 - 8*y^15 + 60*y^14 - 280*y^13 + 1366*y^12 - 4920*y^11 + 18684*y^10 - 54040*y^9 + 169095*y^8 - 396536*y^7 + 1046676*y^6 - 1935192*y^5 + 4326842*y^4 - 5804824*y^3 + 11026220*y^2 - 8393144*y + 13286974, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1366*x^12 - 4920*x^11 + 18684*x^10 - 54040*x^9 + 169095*x^8 - 396536*x^7 + 1046676*x^6 - 1935192*x^5 + 4326842*x^4 - 5804824*x^3 + 11026220*x^2 - 8393144*x + 13286974);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1366*x^12 - 4920*x^11 + 18684*x^10 - 54040*x^9 + 169095*x^8 - 396536*x^7 + 1046676*x^6 - 1935192*x^5 + 4326842*x^4 - 5804824*x^3 + 11026220*x^2 - 8393144*x + 13286974)
 

\( x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1366 x^{12} - 4920 x^{11} + 18684 x^{10} - 54040 x^{9} + \cdots + 13286974 \) Copy content Toggle raw display

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(144621385476274636765576298496\) \(\medspace = 2^{48}\cdot 3^{8}\cdot 23^{8}\) 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:  \(66.45\)
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:  $2^{3}3^{1/2}23^{1/2}\approx 66.4529909033446$
Ramified primes:   \(2\), \(3\), \(23\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1104=2^{4}\cdot 3\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{1104}(1,·)$, $\chi_{1104}(323,·)$, $\chi_{1104}(781,·)$, $\chi_{1104}(1103,·)$, $\chi_{1104}(275,·)$, $\chi_{1104}(277,·)$, $\chi_{1104}(599,·)$, $\chi_{1104}(1057,·)$, $\chi_{1104}(229,·)$, $\chi_{1104}(551,·)$, $\chi_{1104}(553,·)$, $\chi_{1104}(875,·)$, $\chi_{1104}(47,·)$, $\chi_{1104}(505,·)$, $\chi_{1104}(827,·)$, $\chi_{1104}(829,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13\!\cdots\!13}a^{14}-\frac{1}{18\!\cdots\!59}a^{13}+\frac{41\!\cdots\!52}{13\!\cdots\!13}a^{12}+\frac{17\!\cdots\!05}{13\!\cdots\!13}a^{11}+\frac{59\!\cdots\!80}{13\!\cdots\!13}a^{10}-\frac{12\!\cdots\!36}{13\!\cdots\!13}a^{9}-\frac{33\!\cdots\!63}{13\!\cdots\!13}a^{8}-\frac{475795874927790}{18\!\cdots\!59}a^{7}+\frac{40\!\cdots\!85}{13\!\cdots\!13}a^{6}-\frac{17\!\cdots\!64}{13\!\cdots\!13}a^{5}-\frac{24\!\cdots\!24}{13\!\cdots\!13}a^{4}-\frac{49\!\cdots\!82}{13\!\cdots\!13}a^{3}+\frac{44\!\cdots\!34}{13\!\cdots\!13}a^{2}+\frac{26\!\cdots\!62}{13\!\cdots\!13}a+\frac{55\!\cdots\!42}{13\!\cdots\!13}$, $\frac{1}{11\!\cdots\!49}a^{15}+\frac{431129}{11\!\cdots\!49}a^{14}+\frac{45\!\cdots\!90}{11\!\cdots\!49}a^{13}-\frac{71\!\cdots\!98}{16\!\cdots\!07}a^{12}+\frac{18\!\cdots\!42}{11\!\cdots\!49}a^{11}-\frac{33\!\cdots\!02}{11\!\cdots\!49}a^{10}+\frac{74\!\cdots\!37}{16\!\cdots\!07}a^{9}-\frac{12\!\cdots\!89}{24\!\cdots\!67}a^{8}-\frac{63\!\cdots\!42}{11\!\cdots\!49}a^{7}-\frac{28\!\cdots\!18}{16\!\cdots\!07}a^{6}+\frac{39\!\cdots\!65}{11\!\cdots\!49}a^{5}+\frac{37\!\cdots\!57}{11\!\cdots\!49}a^{4}-\frac{48\!\cdots\!69}{11\!\cdots\!49}a^{3}+\frac{32\!\cdots\!55}{11\!\cdots\!49}a^{2}-\frac{10\!\cdots\!09}{11\!\cdots\!49}a-\frac{37\!\cdots\!82}{11\!\cdots\!49}$ 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

$C_{4}\times C_{40}\times C_{120}$, which has order $19200$ (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:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:   $\frac{33216393108}{13\!\cdots\!13}a^{14}-\frac{33216393108}{18\!\cdots\!59}a^{13}+\frac{1839827389234}{13\!\cdots\!13}a^{12}-\frac{8016272562576}{13\!\cdots\!13}a^{11}+\frac{41945512715060}{13\!\cdots\!13}a^{10}-\frac{141786666668538}{13\!\cdots\!13}a^{9}+\frac{590186674346889}{13\!\cdots\!13}a^{8}-\frac{226957972447968}{18\!\cdots\!59}a^{7}+\frac{57\!\cdots\!28}{13\!\cdots\!13}a^{6}-\frac{12\!\cdots\!24}{13\!\cdots\!13}a^{5}+\frac{42\!\cdots\!26}{13\!\cdots\!13}a^{4}-\frac{65\!\cdots\!96}{13\!\cdots\!13}a^{3}+\frac{19\!\cdots\!19}{13\!\cdots\!13}a^{2}-\frac{16\!\cdots\!98}{13\!\cdots\!13}a+\frac{61\!\cdots\!91}{13\!\cdots\!13}$, $\frac{30\!\cdots\!84}{11\!\cdots\!49}a^{15}-\frac{23\!\cdots\!30}{11\!\cdots\!49}a^{14}+\frac{16\!\cdots\!68}{11\!\cdots\!49}a^{13}-\frac{71\!\cdots\!87}{11\!\cdots\!49}a^{12}+\frac{33\!\cdots\!42}{11\!\cdots\!49}a^{11}-\frac{11\!\cdots\!95}{11\!\cdots\!49}a^{10}+\frac{40\!\cdots\!86}{11\!\cdots\!49}a^{9}-\frac{10\!\cdots\!65}{11\!\cdots\!49}a^{8}+\frac{31\!\cdots\!24}{11\!\cdots\!49}a^{7}-\frac{65\!\cdots\!78}{11\!\cdots\!49}a^{6}+\frac{15\!\cdots\!99}{11\!\cdots\!49}a^{5}-\frac{35\!\cdots\!39}{16\!\cdots\!07}a^{4}+\frac{47\!\cdots\!15}{11\!\cdots\!49}a^{3}-\frac{49\!\cdots\!97}{11\!\cdots\!49}a^{2}+\frac{69\!\cdots\!01}{11\!\cdots\!49}a-\frac{27\!\cdots\!47}{11\!\cdots\!49}$, $\frac{41\!\cdots\!48}{11\!\cdots\!49}a^{15}-\frac{31\!\cdots\!60}{11\!\cdots\!49}a^{14}+\frac{22\!\cdots\!34}{11\!\cdots\!49}a^{13}-\frac{99\!\cdots\!11}{11\!\cdots\!49}a^{12}+\frac{46\!\cdots\!06}{11\!\cdots\!49}a^{11}-\frac{15\!\cdots\!24}{11\!\cdots\!49}a^{10}+\frac{58\!\cdots\!95}{11\!\cdots\!49}a^{9}-\frac{15\!\cdots\!89}{11\!\cdots\!49}a^{8}+\frac{46\!\cdots\!12}{11\!\cdots\!49}a^{7}-\frac{10\!\cdots\!72}{11\!\cdots\!49}a^{6}+\frac{24\!\cdots\!93}{11\!\cdots\!49}a^{5}-\frac{56\!\cdots\!15}{16\!\cdots\!07}a^{4}+\frac{80\!\cdots\!04}{11\!\cdots\!49}a^{3}-\frac{85\!\cdots\!84}{11\!\cdots\!49}a^{2}+\frac{12\!\cdots\!15}{11\!\cdots\!49}a-\frac{51\!\cdots\!81}{11\!\cdots\!49}$, $\frac{131874597300868}{13\!\cdots\!63}a^{15}-\frac{989059479756510}{13\!\cdots\!63}a^{14}+\frac{74\!\cdots\!42}{13\!\cdots\!63}a^{13}-\frac{33\!\cdots\!88}{13\!\cdots\!63}a^{12}+\frac{16\!\cdots\!68}{13\!\cdots\!63}a^{11}-\frac{56\!\cdots\!23}{13\!\cdots\!63}a^{10}+\frac{21\!\cdots\!83}{13\!\cdots\!63}a^{9}-\frac{60\!\cdots\!88}{13\!\cdots\!63}a^{8}+\frac{19\!\cdots\!56}{13\!\cdots\!63}a^{7}-\frac{41\!\cdots\!78}{13\!\cdots\!63}a^{6}+\frac{11\!\cdots\!78}{13\!\cdots\!63}a^{5}-\frac{26\!\cdots\!12}{19\!\cdots\!09}a^{4}+\frac{40\!\cdots\!43}{13\!\cdots\!63}a^{3}-\frac{43\!\cdots\!69}{13\!\cdots\!63}a^{2}+\frac{71\!\cdots\!18}{13\!\cdots\!63}a-\frac{14\!\cdots\!95}{13\!\cdots\!63}$, $\frac{208546617962894}{13\!\cdots\!63}a^{15}-\frac{15\!\cdots\!05}{13\!\cdots\!63}a^{14}+\frac{86\!\cdots\!25}{13\!\cdots\!63}a^{13}-\frac{32\!\cdots\!70}{13\!\cdots\!63}a^{12}+\frac{13\!\cdots\!69}{13\!\cdots\!63}a^{11}-\frac{43\!\cdots\!83}{13\!\cdots\!63}a^{10}+\frac{13\!\cdots\!42}{13\!\cdots\!63}a^{9}-\frac{34\!\cdots\!64}{13\!\cdots\!63}a^{8}+\frac{93\!\cdots\!09}{13\!\cdots\!63}a^{7}-\frac{19\!\cdots\!11}{13\!\cdots\!63}a^{6}+\frac{44\!\cdots\!24}{13\!\cdots\!63}a^{5}-\frac{10\!\cdots\!68}{19\!\cdots\!09}a^{4}+\frac{14\!\cdots\!07}{13\!\cdots\!63}a^{3}-\frac{15\!\cdots\!27}{13\!\cdots\!63}a^{2}+\frac{31\!\cdots\!70}{13\!\cdots\!63}a+\frac{68\!\cdots\!11}{13\!\cdots\!63}$, $\frac{15\!\cdots\!66}{11\!\cdots\!49}a^{15}-\frac{23\!\cdots\!13}{11\!\cdots\!49}a^{14}-\frac{29\!\cdots\!91}{11\!\cdots\!49}a^{13}+\frac{22\!\cdots\!03}{11\!\cdots\!49}a^{12}-\frac{98\!\cdots\!49}{11\!\cdots\!49}a^{11}+\frac{56\!\cdots\!02}{11\!\cdots\!49}a^{10}-\frac{18\!\cdots\!50}{11\!\cdots\!49}a^{9}+\frac{75\!\cdots\!21}{11\!\cdots\!49}a^{8}-\frac{18\!\cdots\!25}{11\!\cdots\!49}a^{7}+\frac{59\!\cdots\!44}{11\!\cdots\!49}a^{6}-\frac{11\!\cdots\!46}{11\!\cdots\!49}a^{5}+\frac{42\!\cdots\!61}{16\!\cdots\!07}a^{4}-\frac{40\!\cdots\!47}{11\!\cdots\!49}a^{3}+\frac{86\!\cdots\!89}{11\!\cdots\!49}a^{2}-\frac{61\!\cdots\!57}{11\!\cdots\!49}a+\frac{11\!\cdots\!01}{11\!\cdots\!49}$, $\frac{30\!\cdots\!84}{11\!\cdots\!49}a^{15}-\frac{32\!\cdots\!12}{11\!\cdots\!49}a^{14}+\frac{22\!\cdots\!42}{11\!\cdots\!49}a^{13}-\frac{11\!\cdots\!22}{11\!\cdots\!49}a^{12}+\frac{11\!\cdots\!70}{24\!\cdots\!67}a^{11}-\frac{19\!\cdots\!06}{11\!\cdots\!49}a^{10}+\frac{67\!\cdots\!98}{11\!\cdots\!49}a^{9}-\frac{20\!\cdots\!33}{11\!\cdots\!49}a^{8}+\frac{55\!\cdots\!00}{11\!\cdots\!49}a^{7}-\frac{13\!\cdots\!23}{11\!\cdots\!49}a^{6}+\frac{29\!\cdots\!05}{11\!\cdots\!49}a^{5}-\frac{81\!\cdots\!98}{16\!\cdots\!07}a^{4}+\frac{92\!\cdots\!23}{11\!\cdots\!49}a^{3}-\frac{14\!\cdots\!99}{11\!\cdots\!49}a^{2}+\frac{14\!\cdots\!33}{11\!\cdots\!49}a-\frac{15\!\cdots\!85}{11\!\cdots\!49}$ 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:  \( 11964.310642723332 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 11964.310642723332 \cdot 19200}{2\cdot\sqrt{144621385476274636765576298496}}\cr\approx \mathstrut & 0.733637634017020 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1366*x^12 - 4920*x^11 + 18684*x^10 - 54040*x^9 + 169095*x^8 - 396536*x^7 + 1046676*x^6 - 1935192*x^5 + 4326842*x^4 - 5804824*x^3 + 11026220*x^2 - 8393144*x + 13286974)
 
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^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1366*x^12 - 4920*x^11 + 18684*x^10 - 54040*x^9 + 169095*x^8 - 396536*x^7 + 1046676*x^6 - 1935192*x^5 + 4326842*x^4 - 5804824*x^3 + 11026220*x^2 - 8393144*x + 13286974, 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^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1366*x^12 - 4920*x^11 + 18684*x^10 - 54040*x^9 + 169095*x^8 - 396536*x^7 + 1046676*x^6 - 1935192*x^5 + 4326842*x^4 - 5804824*x^3 + 11026220*x^2 - 8393144*x + 13286974);
 
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^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1366*x^12 - 4920*x^11 + 18684*x^10 - 54040*x^9 + 169095*x^8 - 396536*x^7 + 1046676*x^6 - 1935192*x^5 + 4326842*x^4 - 5804824*x^3 + 11026220*x^2 - 8393144*x + 13286974);
 
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

$C_2^2\times C_4$ (as 16T2):

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 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-138}) \), \(\Q(\sqrt{-46}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-69}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{3}, \sqrt{-46})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-23})\), \(\Q(\sqrt{2}, \sqrt{-69})\), \(\Q(\sqrt{6}, \sqrt{-23})\), \(\Q(\sqrt{2}, \sqrt{-23})\), \(\Q(\sqrt{6}, \sqrt{-46})\), 4.0.9750528.5, 4.0.1083392.5, \(\Q(\zeta_{16})^+\), 4.4.18432.1, 8.0.1485512441856.5, 8.0.380291185115136.58, \(\Q(\zeta_{48})^+\), 8.0.380291185115136.75, 8.0.380291185115136.45, 8.0.95072796278784.65, 8.0.1173738225664.2

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ R ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.1.0.1}{1} }^{16}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.8.24.9$x^{8} + 8 x^{6} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.9$x^{8} + 8 x^{6} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 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}$
\(23\) Copy content Toggle raw display 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}$
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}$