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Properties

Label 18.0.349...283.1
Degree $18$
Signature $[0, 9]$
Discriminant $-3.496\times 10^{54}$
Root discriminant \(1072.01\)
Ramified primes $3,13,61,7069$
Class number $9$ (GRH)
Class group [3, 3] (GRH)
Galois group $S_3\times C_3^3:A_4$ (as 18T347)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 1788*x^15 + 1966668*x^12 - 1619227253*x^9 + 2349501559548*x^6 - 2551860249998748*x^3 + 1705037985527826781)
 
gp: K = bnfinit(y^18 - 1788*y^15 + 1966668*y^12 - 1619227253*y^9 + 2349501559548*y^6 - 2551860249998748*y^3 + 1705037985527826781, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 1788*x^15 + 1966668*x^12 - 1619227253*x^9 + 2349501559548*x^6 - 2551860249998748*x^3 + 1705037985527826781);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 1788*x^15 + 1966668*x^12 - 1619227253*x^9 + 2349501559548*x^6 - 2551860249998748*x^3 + 1705037985527826781)
 

\( x^{18} - 1788 x^{15} + 1966668 x^{12} - 1619227253 x^{9} + 2349501559548 x^{6} + \cdots + 17\!\cdots\!81 \) Copy content Toggle raw display

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

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-3496349243649823208567729584247733921236230435621526283\) \(\medspace = -\,3^{7}\cdot 13^{6}\cdot 61^{12}\cdot 7069^{6}\) 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:  \(1072.01\)
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:  not computed
Ramified primes:   \(3\), \(13\), \(61\), \(7069\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{9}a^{4}-\frac{1}{9}a^{3}-\frac{1}{9}a+\frac{1}{9}$, $\frac{1}{9}a^{5}-\frac{1}{9}a^{3}-\frac{1}{9}a^{2}+\frac{1}{9}$, $\frac{1}{27}a^{6}-\frac{2}{27}a^{3}+\frac{1}{27}$, $\frac{1}{27}a^{7}+\frac{1}{27}a^{4}-\frac{1}{9}a^{3}-\frac{2}{27}a+\frac{1}{9}$, $\frac{1}{81}a^{8}+\frac{1}{81}a^{7}+\frac{1}{81}a^{6}-\frac{2}{81}a^{5}-\frac{2}{81}a^{4}-\frac{2}{81}a^{3}+\frac{1}{81}a^{2}+\frac{1}{81}a+\frac{1}{81}$, $\frac{1}{243}a^{9}-\frac{1}{81}a^{6}-\frac{8}{81}a^{3}+\frac{26}{243}$, $\frac{1}{243}a^{10}-\frac{1}{81}a^{7}+\frac{1}{81}a^{4}-\frac{1}{9}a^{3}-\frac{1}{243}a+\frac{1}{9}$, $\frac{1}{9477}a^{11}+\frac{1}{729}a^{10}+\frac{1}{729}a^{9}+\frac{2}{3159}a^{8}+\frac{2}{243}a^{7}+\frac{2}{243}a^{6}+\frac{148}{3159}a^{5}+\frac{13}{243}a^{4}+\frac{40}{243}a^{3}+\frac{1007}{9477}a^{2}-\frac{46}{729}a-\frac{127}{729}$, $\frac{1}{4229999519733}a^{12}-\frac{5352083068}{4229999519733}a^{9}-\frac{23366913604}{1409999839911}a^{6}+\frac{57761397752}{4229999519733}a^{3}-\frac{1224125}{3540753}$, $\frac{1}{4229999519733}a^{13}-\frac{5352083068}{4229999519733}a^{10}-\frac{23366913604}{1409999839911}a^{7}+\frac{57761397752}{4229999519733}a^{4}-\frac{1224125}{3540753}a$, $\frac{1}{164969981269587}a^{14}+\frac{1}{12689998559199}a^{13}+\frac{1}{12689998559199}a^{12}-\frac{5352083068}{164969981269587}a^{11}-\frac{5352083068}{12689998559199}a^{10}-\frac{5352083068}{12689998559199}a^{9}-\frac{180033562483}{54989993756529}a^{8}-\frac{23366913604}{4229999519733}a^{7}-\frac{23366913604}{4229999519733}a^{6}+\frac{5227760810759}{164969981269587}a^{5}+\frac{57761397752}{12689998559199}a^{4}+\frac{57761397752}{12689998559199}a^{3}-\frac{8699048}{138089367}a^{2}-\frac{4764878}{10622259}a-\frac{4764878}{10622259}$, $\frac{1}{39\!\cdots\!39}a^{15}-\frac{22001497006847}{39\!\cdots\!39}a^{12}+\frac{33\!\cdots\!69}{39\!\cdots\!39}a^{9}+\frac{52\!\cdots\!27}{39\!\cdots\!39}a^{6}+\frac{22\!\cdots\!62}{32\!\cdots\!99}a^{3}-\frac{102239241972130}{273511495387359}$, $\frac{1}{39\!\cdots\!39}a^{16}-\frac{22001497006847}{39\!\cdots\!39}a^{13}+\frac{33\!\cdots\!69}{39\!\cdots\!39}a^{10}+\frac{52\!\cdots\!27}{39\!\cdots\!39}a^{7}-\frac{13\!\cdots\!49}{32\!\cdots\!99}a^{4}+\frac{1}{9}a^{3}-\frac{71849075817979}{273511495387359}a-\frac{1}{9}$, $\frac{1}{15\!\cdots\!21}a^{17}+\frac{1}{11\!\cdots\!17}a^{16}+\frac{1}{11\!\cdots\!17}a^{15}-\frac{22001497006847}{15\!\cdots\!21}a^{14}-\frac{22001497006847}{11\!\cdots\!17}a^{13}-\frac{22001497006847}{11\!\cdots\!17}a^{12}+\frac{33\!\cdots\!69}{15\!\cdots\!21}a^{11}+\frac{33\!\cdots\!69}{11\!\cdots\!17}a^{10}+\frac{33\!\cdots\!69}{11\!\cdots\!17}a^{9}-\frac{67\!\cdots\!58}{15\!\cdots\!21}a^{8}+\frac{19\!\cdots\!84}{11\!\cdots\!17}a^{7}+\frac{19\!\cdots\!84}{11\!\cdots\!17}a^{6}-\frac{50\!\cdots\!66}{12\!\cdots\!61}a^{5}-\frac{13\!\cdots\!12}{98\!\cdots\!97}a^{4}-\frac{13\!\cdots\!12}{98\!\cdots\!97}a^{3}+\frac{12\!\cdots\!80}{10\!\cdots\!01}a^{2}-\frac{92109186587413}{820534486162077}a-\frac{92109186587413}{820534486162077}$ 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

$C_{3}\times C_{3}$, which has order $9$ (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:  $8$
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:   $\frac{7177}{16\!\cdots\!71}a^{15}-\frac{21195103}{16\!\cdots\!71}a^{12}+\frac{9689051104}{56\!\cdots\!57}a^{9}-\frac{18077200577170}{16\!\cdots\!71}a^{6}+\frac{1922907934}{1409999839911}a^{3}+\frac{733129}{393417}$, $\frac{29545}{16\!\cdots\!71}a^{15}-\frac{21765274}{16\!\cdots\!71}a^{12}+\frac{855935164}{56\!\cdots\!57}a^{9}-\frac{23860616169583}{16\!\cdots\!71}a^{6}+\frac{36246438805}{1409999839911}a^{3}-\frac{549890}{393417}$, $\frac{15\!\cdots\!69}{50\!\cdots\!07}a^{17}+\frac{11\!\cdots\!25}{39\!\cdots\!39}a^{16}-\frac{12\!\cdots\!10}{39\!\cdots\!39}a^{15}-\frac{51\!\cdots\!10}{50\!\cdots\!07}a^{14}-\frac{16\!\cdots\!14}{39\!\cdots\!39}a^{13}+\frac{16\!\cdots\!55}{39\!\cdots\!39}a^{12}+\frac{86\!\cdots\!65}{50\!\cdots\!07}a^{11}+\frac{49\!\cdots\!69}{39\!\cdots\!39}a^{10}+\frac{57\!\cdots\!86}{39\!\cdots\!39}a^{9}-\frac{81\!\cdots\!94}{50\!\cdots\!07}a^{8}-\frac{73\!\cdots\!35}{39\!\cdots\!39}a^{7}-\frac{47\!\cdots\!27}{39\!\cdots\!39}a^{6}+\frac{27\!\cdots\!23}{32\!\cdots\!99}a^{5}+\frac{49\!\cdots\!32}{32\!\cdots\!99}a^{4}+\frac{46\!\cdots\!81}{32\!\cdots\!99}a^{3}-\frac{58\!\cdots\!98}{35\!\cdots\!67}a^{2}-\frac{16\!\cdots\!34}{273511495387359}a-\frac{20\!\cdots\!01}{273511495387359}$, $\frac{27\!\cdots\!72}{56\!\cdots\!23}a^{17}+\frac{15\!\cdots\!98}{39\!\cdots\!39}a^{16}+\frac{95\!\cdots\!16}{39\!\cdots\!39}a^{15}-\frac{45\!\cdots\!49}{56\!\cdots\!23}a^{14}-\frac{32\!\cdots\!09}{39\!\cdots\!39}a^{13}-\frac{30\!\cdots\!10}{39\!\cdots\!39}a^{12}+\frac{18\!\cdots\!73}{56\!\cdots\!23}a^{11}+\frac{15\!\cdots\!64}{39\!\cdots\!39}a^{10}+\frac{16\!\cdots\!15}{39\!\cdots\!39}a^{9}-\frac{22\!\cdots\!26}{56\!\cdots\!23}a^{8}-\frac{12\!\cdots\!07}{39\!\cdots\!39}a^{7}-\frac{81\!\cdots\!13}{39\!\cdots\!39}a^{6}+\frac{50\!\cdots\!69}{47\!\cdots\!43}a^{5}+\frac{33\!\cdots\!73}{32\!\cdots\!99}a^{4}+\frac{27\!\cdots\!48}{32\!\cdots\!99}a^{3}-\frac{27\!\cdots\!66}{395072160003963}a^{2}-\frac{22\!\cdots\!06}{273511495387359}a-\frac{23\!\cdots\!42}{273511495387359}$, $\frac{56\!\cdots\!48}{18\!\cdots\!41}a^{17}-\frac{12\!\cdots\!56}{39\!\cdots\!39}a^{16}+\frac{11\!\cdots\!03}{39\!\cdots\!39}a^{15}-\frac{55\!\cdots\!76}{69\!\cdots\!83}a^{14}+\frac{28\!\cdots\!58}{39\!\cdots\!39}a^{13}-\frac{23\!\cdots\!96}{39\!\cdots\!39}a^{12}+\frac{18\!\cdots\!47}{18\!\cdots\!41}a^{11}-\frac{29\!\cdots\!66}{39\!\cdots\!39}a^{10}+\frac{16\!\cdots\!75}{39\!\cdots\!39}a^{9}-\frac{98\!\cdots\!22}{18\!\cdots\!41}a^{8}+\frac{48\!\cdots\!44}{39\!\cdots\!39}a^{7}+\frac{12\!\cdots\!67}{39\!\cdots\!39}a^{6}-\frac{80\!\cdots\!95}{31\!\cdots\!83}a^{5}+\frac{11\!\cdots\!99}{32\!\cdots\!99}a^{4}-\frac{22\!\cdots\!89}{32\!\cdots\!99}a^{3}+\frac{23\!\cdots\!00}{131690720001321}a^{2}-\frac{88\!\cdots\!48}{273511495387359}a+\frac{12\!\cdots\!88}{273511495387359}$, $\frac{21\!\cdots\!57}{50\!\cdots\!07}a^{17}+\frac{10\!\cdots\!20}{13\!\cdots\!13}a^{16}-\frac{63\!\cdots\!79}{42\!\cdots\!87}a^{15}+\frac{18\!\cdots\!39}{50\!\cdots\!07}a^{14}-\frac{17\!\cdots\!74}{13\!\cdots\!13}a^{13}+\frac{56\!\cdots\!07}{42\!\cdots\!87}a^{12}+\frac{23\!\cdots\!67}{50\!\cdots\!07}a^{11}+\frac{80\!\cdots\!54}{13\!\cdots\!13}a^{10}-\frac{55\!\cdots\!92}{42\!\cdots\!87}a^{9}+\frac{18\!\cdots\!01}{50\!\cdots\!07}a^{8}-\frac{16\!\cdots\!19}{13\!\cdots\!13}a^{7}+\frac{55\!\cdots\!83}{42\!\cdots\!87}a^{6}+\frac{27\!\cdots\!71}{42\!\cdots\!87}a^{5}+\frac{10\!\cdots\!80}{10\!\cdots\!33}a^{4}-\frac{67\!\cdots\!96}{35\!\cdots\!67}a^{3}+\frac{15\!\cdots\!55}{35\!\cdots\!67}a^{2}-\frac{18\!\cdots\!34}{91170498462453}a+\frac{59\!\cdots\!98}{273511495387359}$, $\frac{10\!\cdots\!91}{50\!\cdots\!07}a^{17}+\frac{22\!\cdots\!88}{39\!\cdots\!39}a^{16}-\frac{11\!\cdots\!86}{39\!\cdots\!39}a^{15}-\frac{19\!\cdots\!07}{50\!\cdots\!07}a^{14}-\frac{85\!\cdots\!28}{39\!\cdots\!39}a^{13}+\frac{12\!\cdots\!54}{39\!\cdots\!39}a^{12}+\frac{97\!\cdots\!32}{50\!\cdots\!07}a^{11}+\frac{12\!\cdots\!46}{39\!\cdots\!39}a^{10}-\frac{95\!\cdots\!56}{39\!\cdots\!39}a^{9}-\frac{16\!\cdots\!92}{50\!\cdots\!07}a^{8}-\frac{84\!\cdots\!05}{39\!\cdots\!39}a^{7}+\frac{12\!\cdots\!48}{39\!\cdots\!39}a^{6}+\frac{11\!\cdots\!36}{42\!\cdots\!87}a^{5}+\frac{13\!\cdots\!59}{25\!\cdots\!23}a^{4}-\frac{11\!\cdots\!99}{32\!\cdots\!99}a^{3}-\frac{31\!\cdots\!69}{35\!\cdots\!67}a^{2}-\frac{87\!\cdots\!79}{273511495387359}a+\frac{14\!\cdots\!27}{273511495387359}$, $\frac{13\!\cdots\!49}{15\!\cdots\!21}a^{17}+\frac{20\!\cdots\!59}{11\!\cdots\!17}a^{16}-\frac{13\!\cdots\!42}{11\!\cdots\!17}a^{15}-\frac{34\!\cdots\!33}{15\!\cdots\!21}a^{14}-\frac{10\!\cdots\!68}{11\!\cdots\!17}a^{13}+\frac{16\!\cdots\!18}{11\!\cdots\!17}a^{12}+\frac{12\!\cdots\!11}{15\!\cdots\!21}a^{11}+\frac{82\!\cdots\!70}{11\!\cdots\!17}a^{10}-\frac{11\!\cdots\!98}{11\!\cdots\!17}a^{9}-\frac{31\!\cdots\!53}{15\!\cdots\!21}a^{8}-\frac{10\!\cdots\!31}{11\!\cdots\!17}a^{7}+\frac{15\!\cdots\!84}{11\!\cdots\!17}a^{6}+\frac{15\!\cdots\!48}{12\!\cdots\!61}a^{5}+\frac{12\!\cdots\!66}{98\!\cdots\!97}a^{4}-\frac{14\!\cdots\!78}{98\!\cdots\!97}a^{3}-\frac{50\!\cdots\!40}{10\!\cdots\!01}a^{2}-\frac{10\!\cdots\!99}{820534486162077}a+\frac{17\!\cdots\!72}{820534486162077}$ 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:  \( 93871258993300000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 93871258993300000000 \cdot 9}{2\cdot\sqrt{3496349243649823208567729584247733921236230435621526283}}\cr\approx \mathstrut & 3.44791513759044 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 1788*x^15 + 1966668*x^12 - 1619227253*x^9 + 2349501559548*x^6 - 2551860249998748*x^3 + 1705037985527826781)
 
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^18 - 1788*x^15 + 1966668*x^12 - 1619227253*x^9 + 2349501559548*x^6 - 2551860249998748*x^3 + 1705037985527826781, 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^18 - 1788*x^15 + 1966668*x^12 - 1619227253*x^9 + 2349501559548*x^6 - 2551860249998748*x^3 + 1705037985527826781);
 
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^18 - 1788*x^15 + 1966668*x^12 - 1619227253*x^9 + 2349501559548*x^6 - 2551860249998748*x^3 + 1705037985527826781);
 
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

$S_3\times C_3^3:A_4$ (as 18T347):

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 1944
The 39 conjugacy class representatives for $S_3\times C_3^3:A_4$
Character table for $S_3\times C_3^3:A_4$ is not computed

Intermediate fields

3.3.3721.1, 6.0.41537523.1

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

Degree 24 siblings: data not computed
Degree 27 sibling: data not computed
Degree 36 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 $18$ R $18$ ${\href{/padicField/7.3.0.1}{3} }^{6}$ ${\href{/padicField/11.2.0.1}{2} }^{7}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ R $18$ ${\href{/padicField/19.9.0.1}{9} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ $18$ ${\href{/padicField/31.9.0.1}{9} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ $18$

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
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{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}$
\(13\) Copy content Toggle raw display 13.9.6.2$x^{9} - 52 x^{6} + 676 x^{3} + 265837$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
13.9.0.1$x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$$1$$9$$0$$C_9$$[\ ]^{9}$
\(61\) Copy content Toggle raw display 61.3.2.1$x^{3} + 61$$3$$1$$2$$C_3$$[\ ]_{3}$
61.3.2.1$x^{3} + 61$$3$$1$$2$$C_3$$[\ ]_{3}$
61.3.2.1$x^{3} + 61$$3$$1$$2$$C_3$$[\ ]_{3}$
61.3.2.1$x^{3} + 61$$3$$1$$2$$C_3$$[\ ]_{3}$
61.3.2.1$x^{3} + 61$$3$$1$$2$$C_3$$[\ ]_{3}$
61.3.2.1$x^{3} + 61$$3$$1$$2$$C_3$$[\ ]_{3}$
\(7069\) Copy content Toggle raw display Deg $9$$1$$9$$0$$C_9$$[\ ]^{9}$
Deg $9$$3$$3$$6$