Properties

Label 16.0.670...721.1
Degree $16$
Signature $[0, 8]$
Discriminant $6.702\times 10^{35}$
Root discriminant \(173.44\)
Ramified primes $29,83$
Class number $30$ (GRH)
Class group [30] (GRH)
Galois group $\OD_{16}:C_2$ (as 16T36)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 49*x^14 - 154*x^13 + 848*x^12 - 1754*x^11 + 6979*x^10 + 35819*x^9 + 284435*x^8 - 556257*x^7 + 7200689*x^6 - 31438382*x^5 - 11440997*x^4 + 76161103*x^3 + 106278189*x^2 - 64246320*x + 31088575)
 
gp: K = bnfinit(y^16 - 4*y^15 + 49*y^14 - 154*y^13 + 848*y^12 - 1754*y^11 + 6979*y^10 + 35819*y^9 + 284435*y^8 - 556257*y^7 + 7200689*y^6 - 31438382*y^5 - 11440997*y^4 + 76161103*y^3 + 106278189*y^2 - 64246320*y + 31088575, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 49*x^14 - 154*x^13 + 848*x^12 - 1754*x^11 + 6979*x^10 + 35819*x^9 + 284435*x^8 - 556257*x^7 + 7200689*x^6 - 31438382*x^5 - 11440997*x^4 + 76161103*x^3 + 106278189*x^2 - 64246320*x + 31088575);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 49*x^14 - 154*x^13 + 848*x^12 - 1754*x^11 + 6979*x^10 + 35819*x^9 + 284435*x^8 - 556257*x^7 + 7200689*x^6 - 31438382*x^5 - 11440997*x^4 + 76161103*x^3 + 106278189*x^2 - 64246320*x + 31088575)
 

\( x^{16} - 4 x^{15} + 49 x^{14} - 154 x^{13} + 848 x^{12} - 1754 x^{11} + 6979 x^{10} + 35819 x^{9} + 284435 x^{8} - 556257 x^{7} + 7200689 x^{6} - 31438382 x^{5} + \cdots + 31088575 \) 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:   \(670188096064724500217499812793861721\) \(\medspace = 29^{14}\cdot 83^{8}\) 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:  \(173.44\)
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:  $29^{7/8}83^{1/2}\approx 173.4352195192488$
Ramified primes:   \(29\), \(83\) 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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7}a^{7}-\frac{1}{7}a$, $\frac{1}{7}a^{8}-\frac{1}{7}a^{2}$, $\frac{1}{7}a^{9}-\frac{1}{7}a^{3}$, $\frac{1}{7}a^{10}-\frac{1}{7}a^{4}$, $\frac{1}{7}a^{11}-\frac{1}{7}a^{5}$, $\frac{1}{49}a^{12}+\frac{3}{49}a^{11}+\frac{2}{49}a^{10}-\frac{1}{49}a^{9}-\frac{3}{49}a^{8}-\frac{2}{49}a^{7}+\frac{6}{49}a^{6}-\frac{24}{49}a^{5}-\frac{16}{49}a^{4}+\frac{8}{49}a^{3}+\frac{24}{49}a^{2}+\frac{16}{49}a-\frac{2}{7}$, $\frac{1}{245}a^{13}+\frac{2}{245}a^{12}-\frac{1}{245}a^{11}+\frac{11}{245}a^{10}-\frac{9}{245}a^{9}+\frac{3}{49}a^{8}-\frac{13}{245}a^{7}-\frac{79}{245}a^{6}+\frac{57}{245}a^{5}+\frac{108}{245}a^{4}+\frac{23}{245}a^{3}-\frac{24}{49}a^{2}-\frac{58}{245}a-\frac{1}{7}$, $\frac{1}{121755902708755}a^{14}-\frac{119615023182}{121755902708755}a^{13}+\frac{1119061807081}{121755902708755}a^{12}-\frac{1493886611362}{24351180541751}a^{11}+\frac{13493425973}{239206095695}a^{10}+\frac{3284043991066}{121755902708755}a^{9}-\frac{4577724990648}{121755902708755}a^{8}-\frac{5165636889477}{121755902708755}a^{7}-\frac{60843182233072}{121755902708755}a^{6}+\frac{4720472585841}{24351180541751}a^{5}+\frac{43520087800236}{121755902708755}a^{4}-\frac{51642933771197}{121755902708755}a^{3}-\frac{41037158909078}{121755902708755}a^{2}-\frac{38839989511178}{121755902708755}a-\frac{21073538876}{58961696227}$, $\frac{1}{91\!\cdots\!15}a^{15}+\frac{28\!\cdots\!16}{91\!\cdots\!15}a^{14}+\frac{13\!\cdots\!27}{91\!\cdots\!15}a^{13}+\frac{23\!\cdots\!22}{91\!\cdots\!15}a^{12}-\frac{26\!\cdots\!05}{18\!\cdots\!03}a^{11}+\frac{45\!\cdots\!14}{91\!\cdots\!15}a^{10}-\frac{10\!\cdots\!38}{91\!\cdots\!15}a^{9}+\frac{12\!\cdots\!64}{91\!\cdots\!15}a^{8}+\frac{16\!\cdots\!73}{13\!\cdots\!45}a^{7}-\frac{98\!\cdots\!94}{91\!\cdots\!15}a^{6}+\frac{31\!\cdots\!12}{79\!\cdots\!61}a^{5}+\frac{10\!\cdots\!97}{91\!\cdots\!15}a^{4}-\frac{37\!\cdots\!68}{91\!\cdots\!15}a^{3}+\frac{36\!\cdots\!38}{91\!\cdots\!15}a^{2}+\frac{21\!\cdots\!16}{18\!\cdots\!03}a+\frac{25\!\cdots\!08}{44\!\cdots\!31}$ 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:  $7$

Class group and class number

$C_{30}$, which has order $30$ (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:   $\frac{18\!\cdots\!76}{12\!\cdots\!55}a^{15}-\frac{93\!\cdots\!76}{12\!\cdots\!55}a^{14}+\frac{98\!\cdots\!13}{12\!\cdots\!55}a^{13}-\frac{37\!\cdots\!06}{12\!\cdots\!55}a^{12}+\frac{18\!\cdots\!83}{12\!\cdots\!55}a^{11}-\frac{48\!\cdots\!18}{12\!\cdots\!55}a^{10}+\frac{16\!\cdots\!42}{12\!\cdots\!55}a^{9}+\frac{10\!\cdots\!52}{24\!\cdots\!71}a^{8}+\frac{45\!\cdots\!34}{12\!\cdots\!55}a^{7}-\frac{26\!\cdots\!07}{20\!\cdots\!45}a^{6}+\frac{14\!\cdots\!39}{12\!\cdots\!55}a^{5}-\frac{12\!\cdots\!41}{20\!\cdots\!45}a^{4}+\frac{43\!\cdots\!07}{12\!\cdots\!55}a^{3}+\frac{26\!\cdots\!41}{20\!\cdots\!45}a^{2}-\frac{26\!\cdots\!07}{34\!\cdots\!53}a+\frac{95\!\cdots\!49}{83\!\cdots\!81}$, $\frac{20\!\cdots\!84}{18\!\cdots\!03}a^{15}-\frac{44\!\cdots\!01}{91\!\cdots\!15}a^{14}+\frac{17\!\cdots\!69}{91\!\cdots\!15}a^{13}-\frac{14\!\cdots\!62}{91\!\cdots\!15}a^{12}+\frac{40\!\cdots\!93}{91\!\cdots\!15}a^{11}+\frac{17\!\cdots\!31}{18\!\cdots\!03}a^{10}+\frac{42\!\cdots\!21}{91\!\cdots\!15}a^{9}-\frac{40\!\cdots\!17}{91\!\cdots\!15}a^{8}-\frac{92\!\cdots\!31}{18\!\cdots\!35}a^{7}-\frac{15\!\cdots\!11}{91\!\cdots\!15}a^{6}+\frac{17\!\cdots\!38}{39\!\cdots\!05}a^{5}+\frac{68\!\cdots\!03}{18\!\cdots\!03}a^{4}-\frac{78\!\cdots\!72}{91\!\cdots\!15}a^{3}-\frac{14\!\cdots\!07}{91\!\cdots\!15}a^{2}+\frac{85\!\cdots\!67}{91\!\cdots\!15}a-\frac{18\!\cdots\!98}{44\!\cdots\!31}$, $\frac{19\!\cdots\!06}{18\!\cdots\!03}a^{15}-\frac{33\!\cdots\!99}{91\!\cdots\!15}a^{14}+\frac{22\!\cdots\!46}{91\!\cdots\!15}a^{13}-\frac{54\!\cdots\!63}{91\!\cdots\!15}a^{12}+\frac{24\!\cdots\!82}{91\!\cdots\!15}a^{11}-\frac{42\!\cdots\!65}{18\!\cdots\!03}a^{10}+\frac{47\!\cdots\!64}{91\!\cdots\!15}a^{9}+\frac{41\!\cdots\!62}{91\!\cdots\!15}a^{8}+\frac{41\!\cdots\!07}{13\!\cdots\!45}a^{7}-\frac{13\!\cdots\!34}{91\!\cdots\!15}a^{6}-\frac{31\!\cdots\!48}{39\!\cdots\!05}a^{5}+\frac{64\!\cdots\!45}{18\!\cdots\!03}a^{4}+\frac{45\!\cdots\!72}{91\!\cdots\!15}a^{3}-\frac{21\!\cdots\!83}{91\!\cdots\!15}a^{2}+\frac{97\!\cdots\!23}{91\!\cdots\!15}a+\frac{67\!\cdots\!26}{44\!\cdots\!31}$, $\frac{14\!\cdots\!59}{83\!\cdots\!65}a^{15}-\frac{14\!\cdots\!29}{83\!\cdots\!65}a^{14}+\frac{96\!\cdots\!01}{83\!\cdots\!65}a^{13}-\frac{61\!\cdots\!36}{83\!\cdots\!65}a^{12}+\frac{21\!\cdots\!08}{83\!\cdots\!65}a^{11}-\frac{88\!\cdots\!63}{83\!\cdots\!65}a^{10}+\frac{19\!\cdots\!87}{83\!\cdots\!65}a^{9}-\frac{22\!\cdots\!96}{16\!\cdots\!73}a^{8}+\frac{68\!\cdots\!24}{83\!\cdots\!65}a^{7}-\frac{59\!\cdots\!32}{14\!\cdots\!35}a^{6}+\frac{12\!\cdots\!49}{83\!\cdots\!65}a^{5}-\frac{17\!\cdots\!96}{14\!\cdots\!35}a^{4}+\frac{40\!\cdots\!53}{16\!\cdots\!73}a^{3}+\frac{71\!\cdots\!64}{14\!\cdots\!35}a^{2}-\frac{26\!\cdots\!11}{11\!\cdots\!95}a-\frac{20\!\cdots\!25}{82\!\cdots\!29}$, $\frac{65\!\cdots\!57}{91\!\cdots\!15}a^{15}-\frac{36\!\cdots\!00}{18\!\cdots\!03}a^{14}+\frac{28\!\cdots\!22}{91\!\cdots\!15}a^{13}-\frac{12\!\cdots\!51}{18\!\cdots\!03}a^{12}+\frac{43\!\cdots\!21}{91\!\cdots\!15}a^{11}-\frac{53\!\cdots\!37}{91\!\cdots\!15}a^{10}+\frac{32\!\cdots\!91}{91\!\cdots\!15}a^{9}+\frac{28\!\cdots\!74}{91\!\cdots\!15}a^{8}+\frac{21\!\cdots\!54}{91\!\cdots\!15}a^{7}-\frac{23\!\cdots\!44}{18\!\cdots\!03}a^{6}+\frac{18\!\cdots\!96}{39\!\cdots\!05}a^{5}-\frac{14\!\cdots\!76}{91\!\cdots\!15}a^{4}-\frac{61\!\cdots\!20}{18\!\cdots\!03}a^{3}+\frac{31\!\cdots\!62}{91\!\cdots\!15}a^{2}+\frac{12\!\cdots\!69}{91\!\cdots\!15}a+\frac{33\!\cdots\!32}{44\!\cdots\!31}$, $\frac{72\!\cdots\!39}{59\!\cdots\!55}a^{15}-\frac{59\!\cdots\!73}{83\!\cdots\!37}a^{14}+\frac{20\!\cdots\!96}{41\!\cdots\!85}a^{13}-\frac{19\!\cdots\!89}{41\!\cdots\!85}a^{12}+\frac{11\!\cdots\!06}{41\!\cdots\!85}a^{11}-\frac{10\!\cdots\!73}{83\!\cdots\!37}a^{10}+\frac{26\!\cdots\!47}{41\!\cdots\!85}a^{9}-\frac{43\!\cdots\!84}{41\!\cdots\!85}a^{8}+\frac{13\!\cdots\!42}{41\!\cdots\!85}a^{7}+\frac{18\!\cdots\!78}{41\!\cdots\!85}a^{6}-\frac{24\!\cdots\!12}{41\!\cdots\!85}a^{5}-\frac{61\!\cdots\!10}{83\!\cdots\!37}a^{4}+\frac{68\!\cdots\!99}{41\!\cdots\!85}a^{3}+\frac{63\!\cdots\!08}{41\!\cdots\!85}a^{2}-\frac{40\!\cdots\!63}{41\!\cdots\!85}a+\frac{10\!\cdots\!13}{20\!\cdots\!49}$, $\frac{10\!\cdots\!54}{13\!\cdots\!45}a^{15}-\frac{11\!\cdots\!39}{91\!\cdots\!15}a^{14}-\frac{80\!\cdots\!04}{91\!\cdots\!15}a^{13}+\frac{15\!\cdots\!46}{91\!\cdots\!15}a^{12}-\frac{12\!\cdots\!39}{15\!\cdots\!85}a^{11}+\frac{12\!\cdots\!39}{91\!\cdots\!15}a^{10}+\frac{26\!\cdots\!44}{18\!\cdots\!03}a^{9}-\frac{26\!\cdots\!22}{91\!\cdots\!15}a^{8}+\frac{11\!\cdots\!69}{91\!\cdots\!15}a^{7}+\frac{12\!\cdots\!53}{91\!\cdots\!15}a^{6}-\frac{25\!\cdots\!96}{39\!\cdots\!05}a^{5}-\frac{24\!\cdots\!93}{91\!\cdots\!15}a^{4}+\frac{13\!\cdots\!98}{91\!\cdots\!15}a^{3}+\frac{35\!\cdots\!40}{18\!\cdots\!03}a^{2}-\frac{10\!\cdots\!47}{91\!\cdots\!15}a+\frac{25\!\cdots\!19}{44\!\cdots\!31}$ 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:  \( 89005085005.7 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 89005085005.7 \cdot 30}{2\cdot\sqrt{670188096064724500217499812793861721}}\cr\approx \mathstrut & 3.96138151253 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 49*x^14 - 154*x^13 + 848*x^12 - 1754*x^11 + 6979*x^10 + 35819*x^9 + 284435*x^8 - 556257*x^7 + 7200689*x^6 - 31438382*x^5 - 11440997*x^4 + 76161103*x^3 + 106278189*x^2 - 64246320*x + 31088575)
 
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 - 4*x^15 + 49*x^14 - 154*x^13 + 848*x^12 - 1754*x^11 + 6979*x^10 + 35819*x^9 + 284435*x^8 - 556257*x^7 + 7200689*x^6 - 31438382*x^5 - 11440997*x^4 + 76161103*x^3 + 106278189*x^2 - 64246320*x + 31088575, 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 - 4*x^15 + 49*x^14 - 154*x^13 + 848*x^12 - 1754*x^11 + 6979*x^10 + 35819*x^9 + 284435*x^8 - 556257*x^7 + 7200689*x^6 - 31438382*x^5 - 11440997*x^4 + 76161103*x^3 + 106278189*x^2 - 64246320*x + 31088575);
 
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 - 4*x^15 + 49*x^14 - 154*x^13 + 848*x^12 - 1754*x^11 + 6979*x^10 + 35819*x^9 + 284435*x^8 - 556257*x^7 + 7200689*x^6 - 31438382*x^5 - 11440997*x^4 + 76161103*x^3 + 106278189*x^2 - 64246320*x + 31088575);
 
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

$\OD_{16}:C_2$ (as 16T36):

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 32
The 11 conjugacy class representatives for $\OD_{16}:C_2$
Character table for $\OD_{16}:C_2$

Intermediate fields

\(\Q(\sqrt{-83}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{-2407}) \), 4.4.168015821.1, 4.0.24389.1, \(\Q(\sqrt{29}, \sqrt{-83})\), 8.4.818650167082817189.1 x2, 8.0.118834397892701.1 x2, 8.0.28229316106304041.1

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

Galois closure: deg 32
Degree 8 siblings: 8.0.118834397892701.1, 8.4.818650167082817189.1
Degree 16 siblings: 16.0.97283799690045652521048020437489.1, 16.4.670188096064724500217499812793861721.1
Minimal sibling: 8.0.118834397892701.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.2.0.1}{2} }^{8}$ ${\href{/padicField/7.1.0.1}{1} }^{16}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{8}$ R ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.1.0.1}{1} }^{16}$

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
\(29\) Copy content Toggle raw display 29.8.7.2$x^{8} + 29$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.2$x^{8} + 29$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
\(83\) Copy content Toggle raw display 83.2.1.2$x^{2} + 83$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 83$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 83$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 83$$2$$1$$1$$C_2$$[\ ]_{2}$
83.4.2.1$x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.4.2.1$x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$