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Properties

Label 15.1.223...056.1
Degree $15$
Signature $[1, 7]$
Discriminant $-2.231\times 10^{40}$
Root discriminant \(489.66\)
Ramified primes $2,13,61$
Class number $180$ (GRH)
Class group [3, 60] (GRH)
Galois group $D_{15}$ (as 15T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 254*x^13 - 1646*x^12 + 48397*x^11 - 268124*x^10 + 4851472*x^9 - 28394144*x^8 + 351300576*x^7 - 1427698176*x^6 + 7649749760*x^5 - 35334150144*x^4 + 102643653888*x^3 - 184778966016*x^2 + 191206490112*x - 98107490304)
 
gp: K = bnfinit(y^15 - 2*y^14 + 254*y^13 - 1646*y^12 + 48397*y^11 - 268124*y^10 + 4851472*y^9 - 28394144*y^8 + 351300576*y^7 - 1427698176*y^6 + 7649749760*y^5 - 35334150144*y^4 + 102643653888*y^3 - 184778966016*y^2 + 191206490112*y - 98107490304, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 2*x^14 + 254*x^13 - 1646*x^12 + 48397*x^11 - 268124*x^10 + 4851472*x^9 - 28394144*x^8 + 351300576*x^7 - 1427698176*x^6 + 7649749760*x^5 - 35334150144*x^4 + 102643653888*x^3 - 184778966016*x^2 + 191206490112*x - 98107490304);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 2*x^14 + 254*x^13 - 1646*x^12 + 48397*x^11 - 268124*x^10 + 4851472*x^9 - 28394144*x^8 + 351300576*x^7 - 1427698176*x^6 + 7649749760*x^5 - 35334150144*x^4 + 102643653888*x^3 - 184778966016*x^2 + 191206490112*x - 98107490304)
 

\( x^{15} - 2 x^{14} + 254 x^{13} - 1646 x^{12} + 48397 x^{11} - 268124 x^{10} + 4851472 x^{9} + \cdots - 98107490304 \) Copy content Toggle raw display

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

Invariants

Degree:  $15$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-22308540207830291106807065056657300013056\) \(\medspace = -\,2^{14}\cdot 13^{10}\cdot 61^{14}\) 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:  \(489.66\)
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:  $2\cdot 13^{2/3}61^{14/15}\approx 512.8221123219365$
Ramified primes:   \(2\), \(13\), \(61\) 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:  \(\Q(\sqrt{-1}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{3}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{16}a^{7}-\frac{1}{8}a^{5}+\frac{1}{8}a^{4}-\frac{3}{16}a^{3}+\frac{1}{8}a^{2}$, $\frac{1}{192}a^{8}-\frac{1}{96}a^{7}+\frac{5}{96}a^{6}+\frac{5}{96}a^{5}+\frac{37}{192}a^{4}-\frac{5}{48}a^{3}-\frac{17}{48}a^{2}-\frac{5}{12}a$, $\frac{1}{384}a^{9}+\frac{1}{64}a^{7}-\frac{3}{64}a^{6}+\frac{3}{128}a^{5}-\frac{15}{64}a^{4}+\frac{3}{32}a^{3}-\frac{5}{16}a^{2}+\frac{1}{12}a$, $\frac{1}{3072}a^{10}-\frac{1}{768}a^{9}-\frac{1}{1536}a^{8}-\frac{13}{1536}a^{7}-\frac{191}{3072}a^{6}-\frac{103}{1536}a^{5}-\frac{71}{768}a^{4}+\frac{83}{384}a^{3}-\frac{3}{32}a^{2}-\frac{1}{12}a$, $\frac{1}{454656}a^{11}+\frac{23}{227328}a^{10}-\frac{71}{75776}a^{9}-\frac{85}{75776}a^{8}-\frac{2097}{151552}a^{7}+\frac{1507}{37888}a^{6}+\frac{595}{18944}a^{5}-\frac{399}{4736}a^{4}+\frac{4121}{28416}a^{3}-\frac{2761}{7104}a^{2}-\frac{65}{148}a-\frac{1}{2}$, $\frac{1}{10911744}a^{12}+\frac{1}{2727936}a^{11}-\frac{587}{5455872}a^{10}+\frac{2771}{5455872}a^{9}+\frac{3289}{10911744}a^{8}+\frac{57089}{5455872}a^{7}+\frac{2107}{42624}a^{6}+\frac{41521}{681984}a^{5}-\frac{6353}{227328}a^{4}-\frac{23189}{113664}a^{3}-\frac{3815}{85248}a^{2}+\frac{625}{1776}a-\frac{1}{8}$, $\frac{1}{1046741778432}a^{13}-\frac{23431}{523370889216}a^{12}-\frac{412751}{523370889216}a^{11}+\frac{45870425}{523370889216}a^{10}+\frac{762838189}{1046741778432}a^{9}+\frac{21551641}{16355340288}a^{8}+\frac{519644551}{261685444608}a^{7}+\frac{25876573}{1768144896}a^{6}-\frac{496549399}{21807120384}a^{5}-\frac{645775519}{2725890048}a^{4}-\frac{412560391}{16355340288}a^{3}+\frac{139300255}{454315008}a^{2}+\frac{198035}{443667}a+\frac{52155}{127904}$, $\frac{1}{42\!\cdots\!76}a^{14}+\frac{77\!\cdots\!73}{13\!\cdots\!68}a^{13}-\frac{11\!\cdots\!11}{30\!\cdots\!84}a^{12}+\frac{11\!\cdots\!99}{45\!\cdots\!52}a^{11}-\frac{26\!\cdots\!83}{42\!\cdots\!76}a^{10}+\frac{22\!\cdots\!15}{21\!\cdots\!88}a^{9}-\frac{23\!\cdots\!69}{10\!\cdots\!44}a^{8}+\frac{36\!\cdots\!85}{76\!\cdots\!96}a^{7}+\frac{49\!\cdots\!27}{88\!\cdots\!12}a^{6}-\frac{31\!\cdots\!61}{44\!\cdots\!56}a^{5}-\frac{80\!\cdots\!27}{66\!\cdots\!84}a^{4}+\frac{33\!\cdots\!33}{15\!\cdots\!52}a^{3}-\frac{23\!\cdots\!07}{92\!\cdots\!72}a^{2}-\frac{89\!\cdots\!83}{27\!\cdots\!52}a+\frac{48\!\cdots\!93}{28\!\cdots\!04}$ 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:  $2$, $3$

Class group and class number

$C_{3}\times C_{60}$, which has order $180$ (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:   $\frac{10\!\cdots\!73}{15\!\cdots\!88}a^{14}+\frac{10\!\cdots\!95}{55\!\cdots\!08}a^{13}+\frac{95\!\cdots\!23}{55\!\cdots\!08}a^{12}-\frac{25\!\cdots\!55}{82\!\cdots\!24}a^{11}+\frac{32\!\cdots\!83}{11\!\cdots\!16}a^{10}-\frac{12\!\cdots\!41}{34\!\cdots\!88}a^{9}+\frac{75\!\cdots\!97}{27\!\cdots\!04}a^{8}-\frac{35\!\cdots\!33}{69\!\cdots\!76}a^{7}+\frac{40\!\cdots\!27}{23\!\cdots\!92}a^{6}+\frac{41\!\cdots\!83}{28\!\cdots\!24}a^{5}+\frac{44\!\cdots\!87}{17\!\cdots\!44}a^{4}-\frac{86\!\cdots\!25}{14\!\cdots\!12}a^{3}-\frac{50\!\cdots\!51}{81\!\cdots\!12}a^{2}+\frac{55\!\cdots\!27}{16\!\cdots\!08}a-\frac{15\!\cdots\!29}{28\!\cdots\!24}$, $\frac{46\!\cdots\!09}{28\!\cdots\!84}a^{14}+\frac{34\!\cdots\!29}{50\!\cdots\!72}a^{13}+\frac{40\!\cdots\!69}{10\!\cdots\!44}a^{12}-\frac{24\!\cdots\!77}{14\!\cdots\!32}a^{11}+\frac{14\!\cdots\!87}{20\!\cdots\!88}a^{10}-\frac{25\!\cdots\!37}{10\!\cdots\!44}a^{9}+\frac{35\!\cdots\!49}{50\!\cdots\!72}a^{8}-\frac{70\!\cdots\!57}{25\!\cdots\!36}a^{7}+\frac{20\!\cdots\!09}{41\!\cdots\!56}a^{6}-\frac{22\!\cdots\!61}{20\!\cdots\!28}a^{5}+\frac{26\!\cdots\!03}{31\!\cdots\!92}a^{4}-\frac{17\!\cdots\!49}{52\!\cdots\!32}a^{3}+\frac{27\!\cdots\!13}{43\!\cdots\!36}a^{2}-\frac{56\!\cdots\!35}{90\!\cdots\!32}a+\frac{37\!\cdots\!01}{13\!\cdots\!52}$, $\frac{35\!\cdots\!51}{38\!\cdots\!12}a^{14}+\frac{57\!\cdots\!87}{30\!\cdots\!96}a^{13}+\frac{35\!\cdots\!27}{15\!\cdots\!48}a^{12}-\frac{13\!\cdots\!63}{22\!\cdots\!44}a^{11}+\frac{61\!\cdots\!27}{15\!\cdots\!48}a^{10}-\frac{23\!\cdots\!29}{30\!\cdots\!96}a^{9}+\frac{71\!\cdots\!83}{19\!\cdots\!56}a^{8}-\frac{73\!\cdots\!87}{76\!\cdots\!24}a^{7}+\frac{15\!\cdots\!49}{63\!\cdots\!52}a^{6}-\frac{96\!\cdots\!61}{63\!\cdots\!52}a^{5}+\frac{81\!\cdots\!87}{23\!\cdots\!32}a^{4}-\frac{17\!\cdots\!99}{15\!\cdots\!88}a^{3}-\frac{51\!\cdots\!55}{13\!\cdots\!24}a^{2}+\frac{41\!\cdots\!27}{93\!\cdots\!28}a-\frac{10\!\cdots\!43}{17\!\cdots\!72}$, $\frac{12\!\cdots\!63}{21\!\cdots\!88}a^{14}-\frac{33\!\cdots\!57}{38\!\cdots\!48}a^{13}+\frac{15\!\cdots\!53}{10\!\cdots\!44}a^{12}-\frac{14\!\cdots\!77}{15\!\cdots\!32}a^{11}+\frac{85\!\cdots\!93}{30\!\cdots\!84}a^{10}-\frac{15\!\cdots\!87}{10\!\cdots\!44}a^{9}+\frac{14\!\cdots\!77}{53\!\cdots\!72}a^{8}-\frac{40\!\cdots\!91}{26\!\cdots\!36}a^{7}+\frac{12\!\cdots\!87}{63\!\cdots\!08}a^{6}-\frac{16\!\cdots\!31}{22\!\cdots\!28}a^{5}+\frac{12\!\cdots\!31}{33\!\cdots\!92}a^{4}-\frac{10\!\cdots\!43}{55\!\cdots\!32}a^{3}+\frac{21\!\cdots\!51}{46\!\cdots\!36}a^{2}-\frac{62\!\cdots\!59}{96\!\cdots\!32}a+\frac{60\!\cdots\!31}{14\!\cdots\!52}$, $\frac{12\!\cdots\!37}{71\!\cdots\!96}a^{14}-\frac{47\!\cdots\!11}{17\!\cdots\!24}a^{13}+\frac{19\!\cdots\!95}{35\!\cdots\!48}a^{12}-\frac{35\!\cdots\!71}{53\!\cdots\!44}a^{11}+\frac{71\!\cdots\!33}{71\!\cdots\!96}a^{10}-\frac{35\!\cdots\!31}{35\!\cdots\!48}a^{9}+\frac{15\!\cdots\!39}{17\!\cdots\!24}a^{8}-\frac{41\!\cdots\!55}{88\!\cdots\!12}a^{7}+\frac{34\!\cdots\!79}{14\!\cdots\!52}a^{6}-\frac{77\!\cdots\!39}{73\!\cdots\!76}a^{5}+\frac{37\!\cdots\!77}{11\!\cdots\!64}a^{4}-\frac{13\!\cdots\!43}{18\!\cdots\!44}a^{3}+\frac{16\!\cdots\!71}{15\!\cdots\!12}a^{2}-\frac{28\!\cdots\!61}{32\!\cdots\!44}a+\frac{24\!\cdots\!25}{68\!\cdots\!12}$, $\frac{15\!\cdots\!29}{25\!\cdots\!32}a^{14}+\frac{10\!\cdots\!75}{17\!\cdots\!24}a^{13}+\frac{38\!\cdots\!75}{22\!\cdots\!28}a^{12}+\frac{73\!\cdots\!95}{16\!\cdots\!92}a^{11}+\frac{34\!\cdots\!85}{17\!\cdots\!24}a^{10}-\frac{79\!\cdots\!11}{17\!\cdots\!24}a^{9}+\frac{51\!\cdots\!35}{44\!\cdots\!56}a^{8}-\frac{30\!\cdots\!85}{44\!\cdots\!56}a^{7}+\frac{56\!\cdots\!05}{18\!\cdots\!44}a^{6}-\frac{66\!\cdots\!83}{36\!\cdots\!88}a^{5}+\frac{20\!\cdots\!51}{27\!\cdots\!16}a^{4}-\frac{17\!\cdots\!85}{92\!\cdots\!72}a^{3}+\frac{23\!\cdots\!15}{77\!\cdots\!56}a^{2}-\frac{22\!\cdots\!37}{80\!\cdots\!36}a+\frac{28\!\cdots\!51}{24\!\cdots\!92}$, $\frac{90\!\cdots\!95}{53\!\cdots\!72}a^{14}+\frac{61\!\cdots\!51}{53\!\cdots\!72}a^{13}+\frac{20\!\cdots\!67}{66\!\cdots\!84}a^{12}+\frac{11\!\cdots\!19}{99\!\cdots\!52}a^{11}+\frac{19\!\cdots\!85}{53\!\cdots\!72}a^{10}+\frac{21\!\cdots\!37}{53\!\cdots\!72}a^{9}+\frac{67\!\cdots\!43}{19\!\cdots\!24}a^{8}+\frac{55\!\cdots\!91}{13\!\cdots\!68}a^{7}-\frac{82\!\cdots\!69}{55\!\cdots\!32}a^{6}+\frac{48\!\cdots\!61}{11\!\cdots\!64}a^{5}-\frac{33\!\cdots\!09}{11\!\cdots\!64}a^{4}+\frac{24\!\cdots\!39}{27\!\cdots\!16}a^{3}-\frac{36\!\cdots\!85}{23\!\cdots\!68}a^{2}+\frac{39\!\cdots\!49}{24\!\cdots\!08}a-\frac{58\!\cdots\!13}{72\!\cdots\!76}$ 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:  \( 714737732523037.0 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 714737732523037.0 \cdot 180}{2\cdot\sqrt{22308540207830291106807065056657300013056}}\cr\approx \mathstrut & 332.998837501289 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 254*x^13 - 1646*x^12 + 48397*x^11 - 268124*x^10 + 4851472*x^9 - 28394144*x^8 + 351300576*x^7 - 1427698176*x^6 + 7649749760*x^5 - 35334150144*x^4 + 102643653888*x^3 - 184778966016*x^2 + 191206490112*x - 98107490304)
 
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^15 - 2*x^14 + 254*x^13 - 1646*x^12 + 48397*x^11 - 268124*x^10 + 4851472*x^9 - 28394144*x^8 + 351300576*x^7 - 1427698176*x^6 + 7649749760*x^5 - 35334150144*x^4 + 102643653888*x^3 - 184778966016*x^2 + 191206490112*x - 98107490304, 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^15 - 2*x^14 + 254*x^13 - 1646*x^12 + 48397*x^11 - 268124*x^10 + 4851472*x^9 - 28394144*x^8 + 351300576*x^7 - 1427698176*x^6 + 7649749760*x^5 - 35334150144*x^4 + 102643653888*x^3 - 184778966016*x^2 + 191206490112*x - 98107490304);
 
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^15 - 2*x^14 + 254*x^13 - 1646*x^12 + 48397*x^11 - 268124*x^10 + 4851472*x^9 - 28394144*x^8 + 351300576*x^7 - 1427698176*x^6 + 7649749760*x^5 - 35334150144*x^4 + 102643653888*x^3 - 184778966016*x^2 + 191206490112*x - 98107490304);
 
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

$D_{15}$ (as 15T2):

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 30
The 9 conjugacy class representatives for $D_{15}$
Character table for $D_{15}$

Intermediate fields

3.1.2515396.2, 5.1.221533456.1

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 30
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 R ${\href{/padicField/3.2.0.1}{2} }^{7}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.5.0.1}{5} }^{3}$ ${\href{/padicField/7.2.0.1}{2} }^{7}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.2.0.1}{2} }^{7}{,}\,{\href{/padicField/11.1.0.1}{1} }$ R $15$ ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.5.0.1}{5} }^{3}$ ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.1.0.1}{1} }^{15}$ $15$ ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.5.0.1}{5} }^{3}$ ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
\(13\) Copy content Toggle raw display 13.15.10.3$x^{15} + 114244 x^{3} - 4084223$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$
\(61\) Copy content Toggle raw display 61.15.14.5$x^{15} + 427$$15$$1$$14$$C_{15}$$[\ ]_{15}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.2515396.3t2.a.a$2$ $ 2^{2} \cdot 13^{2} \cdot 61^{2}$ 3.1.2515396.2 $S_3$ (as 3T2) $1$ $0$
* 2.14884.5t2.a.a$2$ $ 2^{2} \cdot 61^{2}$ 5.1.221533456.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.14884.5t2.a.b$2$ $ 2^{2} \cdot 61^{2}$ 5.1.221533456.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.2515396.15t2.b.d$2$ $ 2^{2} \cdot 13^{2} \cdot 61^{2}$ 15.1.22308540207830291106807065056657300013056.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2515396.15t2.b.c$2$ $ 2^{2} \cdot 13^{2} \cdot 61^{2}$ 15.1.22308540207830291106807065056657300013056.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2515396.15t2.b.b$2$ $ 2^{2} \cdot 13^{2} \cdot 61^{2}$ 15.1.22308540207830291106807065056657300013056.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2515396.15t2.b.a$2$ $ 2^{2} \cdot 13^{2} \cdot 61^{2}$ 15.1.22308540207830291106807065056657300013056.1 $D_{15}$ (as 15T2) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.