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Properties

Label 18.0.158...559.1
Degree $18$
Signature $[0, 9]$
Discriminant $-1.590\times 10^{29}$
Root discriminant \(41.91\)
Ramified primes $7,13,79$
Class number $127$ (GRH)
Class group [127] (GRH)
Galois group $C_6\times A_4$ (as 18T25)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 7*x^16 - 10*x^15 + 66*x^14 - 104*x^13 + 252*x^12 + 15*x^11 + 1120*x^10 - 603*x^9 + 2562*x^8 + 8*x^7 + 7896*x^6 - 6768*x^5 + 4896*x^4 - 1984*x^3 + 1792*x^2 - 1024*x + 512)
 
gp: K = bnfinit(y^18 - 2*y^17 + 7*y^16 - 10*y^15 + 66*y^14 - 104*y^13 + 252*y^12 + 15*y^11 + 1120*y^10 - 603*y^9 + 2562*y^8 + 8*y^7 + 7896*y^6 - 6768*y^5 + 4896*y^4 - 1984*y^3 + 1792*y^2 - 1024*y + 512, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 + 7*x^16 - 10*x^15 + 66*x^14 - 104*x^13 + 252*x^12 + 15*x^11 + 1120*x^10 - 603*x^9 + 2562*x^8 + 8*x^7 + 7896*x^6 - 6768*x^5 + 4896*x^4 - 1984*x^3 + 1792*x^2 - 1024*x + 512);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 2*x^17 + 7*x^16 - 10*x^15 + 66*x^14 - 104*x^13 + 252*x^12 + 15*x^11 + 1120*x^10 - 603*x^9 + 2562*x^8 + 8*x^7 + 7896*x^6 - 6768*x^5 + 4896*x^4 - 1984*x^3 + 1792*x^2 - 1024*x + 512)
 

\( x^{18} - 2 x^{17} + 7 x^{16} - 10 x^{15} + 66 x^{14} - 104 x^{13} + 252 x^{12} + 15 x^{11} + 1120 x^{10} + \cdots + 512 \) 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:   \(-158992991838916288440704694559\) \(\medspace = -\,7^{12}\cdot 13^{12}\cdot 79^{3}\) 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:  \(41.91\)
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:  $7^{2/3}13^{2/3}79^{1/2}\approx 179.82130310417884$
Ramified primes:   \(7\), \(13\), \(79\) 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:  \(\Q(\sqrt{-79}) \)
$\card{ \Aut(K/\Q) }$:  $6$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{256}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{7}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{8}a^{5}-\frac{1}{4}a^{4}+\frac{1}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{11}-\frac{1}{8}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{16}a^{6}+\frac{3}{8}a^{5}+\frac{1}{16}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{14}-\frac{1}{32}a^{12}-\frac{1}{8}a^{11}+\frac{1}{16}a^{10}+\frac{1}{8}a^{9}+\frac{1}{8}a^{8}+\frac{7}{32}a^{7}+\frac{7}{16}a^{6}+\frac{9}{32}a^{5}-\frac{1}{8}a^{4}+\frac{1}{4}a^{3}$, $\frac{1}{1664}a^{15}+\frac{9}{832}a^{14}+\frac{43}{1664}a^{13}+\frac{1}{832}a^{12}+\frac{87}{832}a^{11}-\frac{7}{104}a^{10}-\frac{35}{416}a^{9}-\frac{337}{1664}a^{8}+\frac{175}{416}a^{7}+\frac{569}{1664}a^{6}-\frac{249}{832}a^{5}+\frac{151}{416}a^{4}-\frac{55}{208}a^{3}+\frac{45}{104}a^{2}-\frac{7}{52}a+\frac{1}{26}$, $\frac{1}{61921929472}a^{16}-\frac{349315}{7740241184}a^{15}+\frac{52463059}{4763225344}a^{14}+\frac{436516393}{15480482368}a^{13}+\frac{1407173485}{30960964736}a^{12}-\frac{1617788331}{15480482368}a^{11}+\frac{1178849285}{15480482368}a^{10}+\frac{4926078023}{61921929472}a^{9}-\frac{5546395677}{30960964736}a^{8}+\frac{30811793697}{61921929472}a^{7}+\frac{1935391585}{15480482368}a^{6}-\frac{1562161149}{3870120592}a^{5}+\frac{393952003}{1935060296}a^{4}-\frac{181241131}{967530148}a^{3}-\frac{273296523}{967530148}a^{2}-\frac{52452077}{483765074}a-\frac{184218193}{483765074}$, $\frac{1}{38\!\cdots\!12}a^{17}+\frac{36781}{19\!\cdots\!56}a^{16}+\frac{763872840347}{38\!\cdots\!12}a^{15}+\frac{130618644389253}{19\!\cdots\!56}a^{14}-\frac{37856894507329}{19\!\cdots\!56}a^{13}+\frac{74331568682889}{23\!\cdots\!32}a^{12}-\frac{601309113466011}{95\!\cdots\!28}a^{11}-\frac{12\!\cdots\!13}{38\!\cdots\!12}a^{10}-\frac{12\!\cdots\!31}{95\!\cdots\!28}a^{9}-\frac{63\!\cdots\!91}{38\!\cdots\!12}a^{8}-\frac{562283423456257}{14\!\cdots\!12}a^{7}-\frac{42\!\cdots\!01}{95\!\cdots\!28}a^{6}+\frac{595446475804059}{47\!\cdots\!64}a^{5}-\frac{358129735575773}{23\!\cdots\!32}a^{4}+\frac{264994040583209}{11\!\cdots\!16}a^{3}+\frac{154907043183825}{597107360717608}a^{2}+\frac{41964274242921}{149276840179402}a+\frac{630889030137}{5741416929977}$ 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

$C_{127}$, which has order $127$ (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:   $\frac{25445000185}{23\!\cdots\!32}a^{17}+\frac{658553548105}{19\!\cdots\!56}a^{16}-\frac{470222661359}{47\!\cdots\!64}a^{15}+\frac{4355524315839}{19\!\cdots\!56}a^{14}-\frac{237538928021}{298553680358804}a^{13}+\frac{17638213096533}{95\!\cdots\!28}a^{12}-\frac{17118300107273}{47\!\cdots\!64}a^{11}+\frac{17143845753507}{47\!\cdots\!64}a^{10}-\frac{165763541393377}{19\!\cdots\!56}a^{9}+\frac{117366835988829}{95\!\cdots\!28}a^{8}-\frac{9017363973067}{14\!\cdots\!12}a^{7}+\frac{9082384375279}{183725341759264}a^{6}-\frac{51323123036941}{11\!\cdots\!16}a^{5}+\frac{17440762716755}{597107360717608}a^{4}-\frac{7633225360031}{597107360717608}a^{3}+\frac{3262479902333}{298553680358804}a^{2}-\frac{36483182052}{5741416929977}a-\frac{65998978308227}{149276840179402}$, $\frac{25445000185}{23\!\cdots\!32}a^{17}-\frac{658553548105}{19\!\cdots\!56}a^{16}+\frac{470222661359}{47\!\cdots\!64}a^{15}-\frac{4355524315839}{19\!\cdots\!56}a^{14}+\frac{237538928021}{298553680358804}a^{13}-\frac{17638213096533}{95\!\cdots\!28}a^{12}+\frac{17118300107273}{47\!\cdots\!64}a^{11}-\frac{17143845753507}{47\!\cdots\!64}a^{10}+\frac{165763541393377}{19\!\cdots\!56}a^{9}-\frac{117366835988829}{95\!\cdots\!28}a^{8}+\frac{9017363973067}{14\!\cdots\!12}a^{7}-\frac{9082384375279}{183725341759264}a^{6}+\frac{51323123036941}{11\!\cdots\!16}a^{5}-\frac{17440762716755}{597107360717608}a^{4}+\frac{7633225360031}{597107360717608}a^{3}-\frac{3262479902333}{298553680358804}a^{2}+\frac{36483182052}{5741416929977}a-\frac{83277861871175}{149276840179402}$, $\frac{3688019}{1190806336}a^{17}+\frac{280896435}{61921929472}a^{16}-\frac{36031399}{1935060296}a^{15}+\frac{1258349197}{61921929472}a^{14}-\frac{1471105235}{7740241184}a^{13}+\frac{6739635875}{30960964736}a^{12}-\frac{9697584799}{15480482368}a^{11}-\frac{245407057}{595403168}a^{10}-\frac{221298075667}{61921929472}a^{9}+\frac{193540297}{2381612672}a^{8}-\frac{439973659197}{61921929472}a^{7}-\frac{27948281359}{7740241184}a^{6}-\frac{191812348283}{7740241184}a^{5}+\frac{17255520163}{1935060296}a^{4}-\frac{9248633165}{1935060296}a^{3}+\frac{3574184139}{967530148}a^{2}-\frac{1221421113}{483765074}a+\frac{1167406839}{483765074}$, $\frac{16683339}{9526450688}a^{17}+\frac{217962333}{61921929472}a^{16}-\frac{1533536845}{123843858944}a^{15}+\frac{1092996805}{61921929472}a^{14}-\frac{7173354629}{61921929472}a^{13}+\frac{1415010829}{7740241184}a^{12}-\frac{13803581151}{30960964736}a^{11}-\frac{233043301}{9526450688}a^{10}-\frac{60458781499}{30960964736}a^{9}+\frac{9723221669}{9526450688}a^{8}-\frac{281103589717}{61921929472}a^{7}-\frac{2257959035}{30960964736}a^{6}-\frac{211474312701}{15480482368}a^{5}+\frac{90599697845}{7740241184}a^{4}-\frac{32882032743}{3870120592}a^{3}+\frac{6636570963}{1935060296}a^{2}-\frac{206973058}{241882537}a+\frac{186610114}{241882537}$, $\frac{22700703820617}{47\!\cdots\!64}a^{17}+\frac{23360377434891}{47\!\cdots\!64}a^{16}-\frac{235552439095781}{95\!\cdots\!28}a^{15}+\frac{412593783957}{22965667719908}a^{14}-\frac{26\!\cdots\!45}{95\!\cdots\!28}a^{13}+\frac{975707872077925}{47\!\cdots\!64}a^{12}-\frac{36\!\cdots\!57}{47\!\cdots\!64}a^{11}-\frac{51\!\cdots\!87}{47\!\cdots\!64}a^{10}-\frac{26\!\cdots\!95}{47\!\cdots\!64}a^{9}-\frac{19\!\cdots\!63}{95\!\cdots\!28}a^{8}-\frac{46\!\cdots\!01}{47\!\cdots\!64}a^{7}-\frac{94\!\cdots\!03}{95\!\cdots\!28}a^{6}-\frac{18\!\cdots\!59}{47\!\cdots\!64}a^{5}-\frac{30\!\cdots\!51}{23\!\cdots\!32}a^{4}+\frac{67\!\cdots\!05}{11\!\cdots\!16}a^{3}+\frac{15\!\cdots\!13}{597107360717608}a^{2}-\frac{15\!\cdots\!87}{298553680358804}a-\frac{338163110622953}{149276840179402}$, $\frac{45403655501441}{95\!\cdots\!28}a^{17}-\frac{13830388516509}{23\!\cdots\!32}a^{16}+\frac{63061855308157}{23\!\cdots\!32}a^{15}-\frac{8687957285017}{367450683518528}a^{14}+\frac{26\!\cdots\!33}{95\!\cdots\!28}a^{13}-\frac{12\!\cdots\!67}{47\!\cdots\!64}a^{12}+\frac{40\!\cdots\!29}{47\!\cdots\!64}a^{11}+\frac{84\!\cdots\!75}{95\!\cdots\!28}a^{10}+\frac{26\!\cdots\!51}{47\!\cdots\!64}a^{9}+\frac{26\!\cdots\!47}{23\!\cdots\!32}a^{8}+\frac{12\!\cdots\!29}{11\!\cdots\!16}a^{7}+\frac{76\!\cdots\!37}{95\!\cdots\!28}a^{6}+\frac{18\!\cdots\!21}{47\!\cdots\!64}a^{5}-\frac{998707801248511}{183725341759264}a^{4}+\frac{180596818035551}{11\!\cdots\!16}a^{3}-\frac{24\!\cdots\!25}{597107360717608}a^{2}+\frac{13\!\cdots\!47}{298553680358804}a+\frac{88207161255321}{149276840179402}$, $\frac{146951609549557}{38\!\cdots\!12}a^{17}+\frac{45015209002535}{95\!\cdots\!28}a^{16}-\frac{814889448656899}{38\!\cdots\!12}a^{15}+\frac{92339517133235}{47\!\cdots\!64}a^{14}-\frac{43\!\cdots\!49}{19\!\cdots\!56}a^{13}+\frac{20\!\cdots\!37}{95\!\cdots\!28}a^{12}-\frac{65\!\cdots\!61}{95\!\cdots\!28}a^{11}-\frac{27\!\cdots\!67}{38\!\cdots\!12}a^{10}-\frac{85\!\cdots\!93}{19\!\cdots\!56}a^{9}-\frac{32\!\cdots\!77}{38\!\cdots\!12}a^{8}-\frac{617786668495869}{74638420089701}a^{7}-\frac{578437345348389}{91862670879632}a^{6}-\frac{74\!\cdots\!97}{23\!\cdots\!32}a^{5}+\frac{53\!\cdots\!75}{11\!\cdots\!16}a^{4}-\frac{3481683864543}{22965667719908}a^{3}+\frac{972606886916477}{298553680358804}a^{2}-\frac{11\!\cdots\!95}{298553680358804}a-\frac{109269226027974}{74638420089701}$, $\frac{3669069641251}{14\!\cdots\!12}a^{17}+\frac{77292609363637}{19\!\cdots\!56}a^{16}-\frac{300134426551815}{19\!\cdots\!56}a^{15}+\frac{359577342926291}{19\!\cdots\!56}a^{14}-\frac{747461049345479}{47\!\cdots\!64}a^{13}+\frac{19\!\cdots\!01}{95\!\cdots\!28}a^{12}-\frac{25\!\cdots\!23}{47\!\cdots\!64}a^{11}-\frac{47\!\cdots\!89}{19\!\cdots\!56}a^{10}-\frac{54\!\cdots\!99}{19\!\cdots\!56}a^{9}+\frac{94\!\cdots\!05}{19\!\cdots\!56}a^{8}-\frac{11\!\cdots\!83}{19\!\cdots\!56}a^{7}-\frac{20\!\cdots\!93}{95\!\cdots\!28}a^{6}-\frac{94\!\cdots\!29}{47\!\cdots\!64}a^{5}+\frac{24\!\cdots\!43}{23\!\cdots\!32}a^{4}-\frac{76\!\cdots\!15}{11\!\cdots\!16}a^{3}+\frac{163517987096315}{45931335439816}a^{2}-\frac{529434182201573}{298553680358804}a+\frac{102387026583339}{74638420089701}$ 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:  \( 205236.82590785183 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 205236.82590785183 \cdot 127}{2\cdot\sqrt{158992991838916288440704694559}}\cr\approx \mathstrut & 0.498837207190409 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 7*x^16 - 10*x^15 + 66*x^14 - 104*x^13 + 252*x^12 + 15*x^11 + 1120*x^10 - 603*x^9 + 2562*x^8 + 8*x^7 + 7896*x^6 - 6768*x^5 + 4896*x^4 - 1984*x^3 + 1792*x^2 - 1024*x + 512)
 
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 - 2*x^17 + 7*x^16 - 10*x^15 + 66*x^14 - 104*x^13 + 252*x^12 + 15*x^11 + 1120*x^10 - 603*x^9 + 2562*x^8 + 8*x^7 + 7896*x^6 - 6768*x^5 + 4896*x^4 - 1984*x^3 + 1792*x^2 - 1024*x + 512, 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 - 2*x^17 + 7*x^16 - 10*x^15 + 66*x^14 - 104*x^13 + 252*x^12 + 15*x^11 + 1120*x^10 - 603*x^9 + 2562*x^8 + 8*x^7 + 7896*x^6 - 6768*x^5 + 4896*x^4 - 1984*x^3 + 1792*x^2 - 1024*x + 512);
 
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 - 2*x^17 + 7*x^16 - 10*x^15 + 66*x^14 - 104*x^13 + 252*x^12 + 15*x^11 + 1120*x^10 - 603*x^9 + 2562*x^8 + 8*x^7 + 7896*x^6 - 6768*x^5 + 4896*x^4 - 1984*x^3 + 1792*x^2 - 1024*x + 512);
 
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

$C_6\times A_4$ (as 18T25):

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 72
The 24 conjugacy class representatives for $C_6\times A_4$
Character table for $C_6\times A_4$

Intermediate fields

3.3.169.1, \(\Q(\zeta_{7})^+\), 3.3.8281.2, 3.3.8281.1, 6.0.2256319.1, 9.9.567869252041.1

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 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 ${\href{/padicField/2.3.0.1}{3} }^{6}$ ${\href{/padicField/3.6.0.1}{6} }^{3}$ ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ R ${\href{/padicField/11.3.0.1}{3} }^{6}$ R ${\href{/padicField/17.6.0.1}{6} }^{3}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.3.0.1}{3} }^{6}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }^{3}$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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
\(7\) Copy content Toggle raw display 7.18.12.1$x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$$3$$6$$12$$C_6 \times C_3$$[\ ]_{3}^{6}$
\(13\) Copy content Toggle raw display 13.3.2.2$x^{3} + 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} + 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} + 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} + 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} + 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} + 13$$3$$1$$2$$C_3$$[\ ]_{3}$
\(79\) Copy content Toggle raw display 79.3.0.1$x^{3} + 9 x + 76$$1$$3$$0$$C_3$$[\ ]^{3}$
79.3.0.1$x^{3} + 9 x + 76$$1$$3$$0$$C_3$$[\ ]^{3}$
79.6.0.1$x^{6} + 19 x^{3} + 28 x^{2} + 68 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
79.6.3.1$x^{6} + 56169 x^{2} - 37470964$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$