Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 71*x^13 + 708*x^12 - 9630*x^11 + 133026*x^10 - 824449*x^9 + 4611324*x^8 - 10911485*x^7 - 723492*x^6 + 271409820*x^5 - 1328274288*x^4 + 3021383088*x^3 + 24795072*x^2 - 36616748544*x + 88119373824)
gp: K = bnfinit(y^15 - 6*y^14 + 71*y^13 + 708*y^12 - 9630*y^11 + 133026*y^10 - 824449*y^9 + 4611324*y^8 - 10911485*y^7 - 723492*y^6 + 271409820*y^5 - 1328274288*y^4 + 3021383088*y^3 + 24795072*y^2 - 36616748544*y + 88119373824, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 6*x^14 + 71*x^13 + 708*x^12 - 9630*x^11 + 133026*x^10 - 824449*x^9 + 4611324*x^8 - 10911485*x^7 - 723492*x^6 + 271409820*x^5 - 1328274288*x^4 + 3021383088*x^3 + 24795072*x^2 - 36616748544*x + 88119373824);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 6*x^14 + 71*x^13 + 708*x^12 - 9630*x^11 + 133026*x^10 - 824449*x^9 + 4611324*x^8 - 10911485*x^7 - 723492*x^6 + 271409820*x^5 - 1328274288*x^4 + 3021383088*x^3 + 24795072*x^2 - 36616748544*x + 88119373824)
\( x^{15} - 6 x^{14} + 71 x^{13} + 708 x^{12} - 9630 x^{11} + 133026 x^{10} - 824449 x^{9} + \cdots + 88119373824 \)
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: |
\(-25201062723784606393449803925478522347\)
\(\medspace = -\,3^{7}\cdot 271^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(311.48\) | 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: | $3^{1/2}271^{14/15}\approx 323.0962857686453$ | ||
| Ramified primes: |
\(3\), \(271\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{6}+\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{3}a^{2}-\frac{1}{2}a$, $\frac{1}{12}a^{7}-\frac{1}{12}a^{6}+\frac{1}{12}a^{5}-\frac{1}{12}a^{4}+\frac{1}{12}a^{3}-\frac{1}{12}a^{2}$, $\frac{1}{36}a^{8}+\frac{1}{36}a^{7}+\frac{1}{36}a^{6}+\frac{7}{36}a^{5}-\frac{5}{36}a^{4}-\frac{17}{36}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{144}a^{9}-\frac{1}{72}a^{8}-\frac{1}{72}a^{7}-\frac{1}{18}a^{6}+\frac{7}{36}a^{5}+\frac{11}{72}a^{4}-\frac{23}{48}a^{3}-\frac{3}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{288}a^{10}-\frac{1}{144}a^{8}-\frac{1}{36}a^{7}+\frac{1}{18}a^{6}+\frac{17}{144}a^{5}-\frac{5}{32}a^{4}-\frac{29}{72}a^{3}+\frac{1}{3}a^{2}-\frac{1}{12}a$, $\frac{1}{7488}a^{11}+\frac{1}{936}a^{10}-\frac{1}{1248}a^{9}-\frac{1}{117}a^{8}+\frac{19}{936}a^{7}-\frac{239}{3744}a^{6}-\frac{557}{7488}a^{5}+\frac{113}{624}a^{4}-\frac{523}{1872}a^{3}+\frac{1}{156}a^{2}-\frac{1}{156}a+\frac{3}{13}$, $\frac{1}{411840}a^{12}+\frac{1}{68640}a^{11}-\frac{23}{41184}a^{10}-\frac{1}{1320}a^{9}+\frac{71}{6435}a^{8}+\frac{785}{41184}a^{7}-\frac{15617}{411840}a^{6}+\frac{1567}{15840}a^{5}-\frac{1795}{20592}a^{4}-\frac{13021}{102960}a^{3}-\frac{5323}{17160}a^{2}-\frac{593}{1716}a+\frac{202}{715}$, $\frac{1}{192741120}a^{13}+\frac{61}{64247040}a^{12}+\frac{5971}{96370560}a^{11}+\frac{48493}{32123520}a^{10}+\frac{413}{267696}a^{9}+\frac{163663}{32123520}a^{8}-\frac{529579}{14826240}a^{7}+\frac{1482331}{64247040}a^{6}-\frac{6049499}{48185280}a^{5}+\frac{423641}{5353920}a^{4}-\frac{14264}{50193}a^{3}+\frac{5761}{13520}a^{2}-\frac{148073}{334620}a-\frac{4516}{9295}$, $\frac{1}{88\!\cdots\!40}a^{14}+\frac{38\!\cdots\!27}{14\!\cdots\!84}a^{13}-\frac{13\!\cdots\!79}{21\!\cdots\!40}a^{12}+\frac{11\!\cdots\!23}{33\!\cdots\!36}a^{11}+\frac{38\!\cdots\!77}{44\!\cdots\!80}a^{10}+\frac{56\!\cdots\!67}{14\!\cdots\!40}a^{9}+\frac{16\!\cdots\!51}{88\!\cdots\!40}a^{8}-\frac{11\!\cdots\!99}{51\!\cdots\!60}a^{7}-\frac{40\!\cdots\!71}{88\!\cdots\!40}a^{6}+\frac{12\!\cdots\!01}{49\!\cdots\!28}a^{5}-\frac{44\!\cdots\!69}{74\!\cdots\!20}a^{4}+\frac{25\!\cdots\!77}{20\!\cdots\!20}a^{3}+\frac{15\!\cdots\!97}{61\!\cdots\!60}a^{2}+\frac{77\!\cdots\!37}{17\!\cdots\!60}a-\frac{40\!\cdots\!72}{23\!\cdots\!55}$
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
$C_{3}\times C_{3}\times C_{45}$, which has order $405$ (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 $2$)
| 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: |
$\frac{99\!\cdots\!79}{72\!\cdots\!32}a^{14}-\frac{47\!\cdots\!33}{36\!\cdots\!16}a^{13}+\frac{17\!\cdots\!61}{19\!\cdots\!36}a^{12}+\frac{48\!\cdots\!11}{54\!\cdots\!24}a^{11}-\frac{19\!\cdots\!39}{10\!\cdots\!48}a^{10}+\frac{21\!\cdots\!89}{10\!\cdots\!48}a^{9}-\frac{28\!\cdots\!41}{21\!\cdots\!96}a^{8}+\frac{10\!\cdots\!87}{16\!\cdots\!96}a^{7}-\frac{29\!\cdots\!17}{21\!\cdots\!96}a^{6}-\frac{14\!\cdots\!27}{60\!\cdots\!36}a^{5}+\frac{16\!\cdots\!59}{54\!\cdots\!24}a^{4}-\frac{17\!\cdots\!81}{15\!\cdots\!84}a^{3}+\frac{16\!\cdots\!47}{15\!\cdots\!84}a^{2}+\frac{23\!\cdots\!21}{37\!\cdots\!96}a-\frac{66\!\cdots\!03}{46\!\cdots\!37}$, $\frac{60\!\cdots\!91}{17\!\cdots\!20}a^{14}-\frac{11\!\cdots\!23}{29\!\cdots\!20}a^{13}+\frac{95\!\cdots\!61}{42\!\cdots\!20}a^{12}+\frac{31\!\cdots\!59}{14\!\cdots\!60}a^{11}-\frac{50\!\cdots\!61}{97\!\cdots\!40}a^{10}+\frac{15\!\cdots\!49}{29\!\cdots\!20}a^{9}-\frac{63\!\cdots\!07}{17\!\cdots\!20}a^{8}+\frac{13\!\cdots\!15}{91\!\cdots\!48}a^{7}-\frac{42\!\cdots\!23}{17\!\cdots\!20}a^{6}-\frac{87\!\cdots\!49}{48\!\cdots\!20}a^{5}+\frac{18\!\cdots\!27}{14\!\cdots\!60}a^{4}-\frac{17\!\cdots\!81}{40\!\cdots\!60}a^{3}+\frac{33\!\cdots\!33}{12\!\cdots\!80}a^{2}+\frac{14\!\cdots\!63}{33\!\cdots\!80}a-\frac{27\!\cdots\!15}{28\!\cdots\!59}$, $\frac{31\!\cdots\!09}{26\!\cdots\!80}a^{14}-\frac{18\!\cdots\!83}{79\!\cdots\!60}a^{13}+\frac{45\!\cdots\!13}{40\!\cdots\!80}a^{12}+\frac{15\!\cdots\!13}{43\!\cdots\!80}a^{11}-\frac{19\!\cdots\!93}{73\!\cdots\!80}a^{10}+\frac{11\!\cdots\!11}{43\!\cdots\!80}a^{9}-\frac{57\!\cdots\!59}{26\!\cdots\!80}a^{8}+\frac{27\!\cdots\!97}{30\!\cdots\!40}a^{7}-\frac{30\!\cdots\!39}{13\!\cdots\!40}a^{6}-\frac{15\!\cdots\!91}{18\!\cdots\!20}a^{5}+\frac{19\!\cdots\!69}{27\!\cdots\!80}a^{4}-\frac{86\!\cdots\!17}{30\!\cdots\!20}a^{3}+\frac{31\!\cdots\!57}{91\!\cdots\!60}a^{2}+\frac{28\!\cdots\!19}{11\!\cdots\!05}a-\frac{94\!\cdots\!17}{14\!\cdots\!95}$, $\frac{48\!\cdots\!49}{29\!\cdots\!80}a^{14}-\frac{17\!\cdots\!35}{37\!\cdots\!56}a^{13}+\frac{69\!\cdots\!53}{72\!\cdots\!80}a^{12}+\frac{89\!\cdots\!53}{61\!\cdots\!60}a^{11}-\frac{18\!\cdots\!97}{16\!\cdots\!60}a^{10}+\frac{17\!\cdots\!23}{98\!\cdots\!56}a^{9}-\frac{45\!\cdots\!57}{59\!\cdots\!36}a^{8}+\frac{75\!\cdots\!99}{15\!\cdots\!20}a^{7}-\frac{92\!\cdots\!47}{29\!\cdots\!80}a^{6}-\frac{60\!\cdots\!49}{57\!\cdots\!60}a^{5}+\frac{93\!\cdots\!59}{24\!\cdots\!40}a^{4}-\frac{46\!\cdots\!87}{45\!\cdots\!16}a^{3}+\frac{30\!\cdots\!71}{18\!\cdots\!20}a^{2}+\frac{72\!\cdots\!41}{19\!\cdots\!40}a-\frac{10\!\cdots\!81}{21\!\cdots\!05}$, $\frac{27\!\cdots\!09}{88\!\cdots\!40}a^{14}-\frac{63\!\cdots\!97}{74\!\cdots\!20}a^{13}+\frac{21\!\cdots\!97}{16\!\cdots\!80}a^{12}+\frac{97\!\cdots\!79}{37\!\cdots\!60}a^{11}-\frac{79\!\cdots\!01}{49\!\cdots\!80}a^{10}+\frac{41\!\cdots\!11}{17\!\cdots\!72}a^{9}-\frac{76\!\cdots\!37}{88\!\cdots\!40}a^{8}+\frac{26\!\cdots\!61}{51\!\cdots\!60}a^{7}+\frac{71\!\cdots\!61}{88\!\cdots\!40}a^{6}-\frac{65\!\cdots\!93}{24\!\cdots\!40}a^{5}+\frac{29\!\cdots\!87}{74\!\cdots\!20}a^{4}-\frac{43\!\cdots\!65}{41\!\cdots\!44}a^{3}+\frac{81\!\cdots\!13}{12\!\cdots\!32}a^{2}+\frac{47\!\cdots\!87}{13\!\cdots\!20}a-\frac{10\!\cdots\!09}{21\!\cdots\!05}$, $\frac{30\!\cdots\!91}{88\!\cdots\!40}a^{14}-\frac{89\!\cdots\!01}{14\!\cdots\!84}a^{13}+\frac{61\!\cdots\!63}{21\!\cdots\!40}a^{12}-\frac{40\!\cdots\!81}{18\!\cdots\!80}a^{11}-\frac{24\!\cdots\!93}{44\!\cdots\!80}a^{10}+\frac{85\!\cdots\!13}{14\!\cdots\!40}a^{9}-\frac{51\!\cdots\!11}{88\!\cdots\!40}a^{8}+\frac{13\!\cdots\!11}{51\!\cdots\!60}a^{7}-\frac{14\!\cdots\!61}{17\!\cdots\!08}a^{6}-\frac{13\!\cdots\!57}{74\!\cdots\!80}a^{5}+\frac{15\!\cdots\!21}{74\!\cdots\!20}a^{4}-\frac{59\!\cdots\!59}{68\!\cdots\!40}a^{3}+\frac{48\!\cdots\!87}{47\!\cdots\!20}a^{2}+\frac{12\!\cdots\!87}{17\!\cdots\!60}a-\frac{45\!\cdots\!84}{23\!\cdots\!55}$, $\frac{85\!\cdots\!57}{88\!\cdots\!40}a^{14}+\frac{16\!\cdots\!17}{18\!\cdots\!48}a^{13}+\frac{36\!\cdots\!69}{21\!\cdots\!40}a^{12}+\frac{10\!\cdots\!09}{11\!\cdots\!30}a^{11}+\frac{73\!\cdots\!79}{49\!\cdots\!80}a^{10}+\frac{55\!\cdots\!31}{29\!\cdots\!68}a^{9}+\frac{51\!\cdots\!43}{17\!\cdots\!08}a^{8}-\frac{51\!\cdots\!13}{51\!\cdots\!60}a^{7}+\frac{54\!\cdots\!09}{88\!\cdots\!40}a^{6}+\frac{64\!\cdots\!99}{24\!\cdots\!40}a^{5}-\frac{65\!\cdots\!73}{74\!\cdots\!20}a^{4}+\frac{15\!\cdots\!37}{41\!\cdots\!44}a^{3}-\frac{52\!\cdots\!79}{47\!\cdots\!20}a^{2}-\frac{44\!\cdots\!91}{17\!\cdots\!60}a+\frac{31\!\cdots\!14}{23\!\cdots\!55}$
| 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: | \( 4216988041825.0723 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 4216988041825.0723 \cdot 405}{2\cdot\sqrt{25201062723784606393449803925478522347}}\cr\approx \mathstrut & 131.524616321235 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 71*x^13 + 708*x^12 - 9630*x^11 + 133026*x^10 - 824449*x^9 + 4611324*x^8 - 10911485*x^7 - 723492*x^6 + 271409820*x^5 - 1328274288*x^4 + 3021383088*x^3 + 24795072*x^2 - 36616748544*x + 88119373824)
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 - 6*x^14 + 71*x^13 + 708*x^12 - 9630*x^11 + 133026*x^10 - 824449*x^9 + 4611324*x^8 - 10911485*x^7 - 723492*x^6 + 271409820*x^5 - 1328274288*x^4 + 3021383088*x^3 + 24795072*x^2 - 36616748544*x + 88119373824, 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 - 6*x^14 + 71*x^13 + 708*x^12 - 9630*x^11 + 133026*x^10 - 824449*x^9 + 4611324*x^8 - 10911485*x^7 - 723492*x^6 + 271409820*x^5 - 1328274288*x^4 + 3021383088*x^3 + 24795072*x^2 - 36616748544*x + 88119373824);
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 - 6*x^14 + 71*x^13 + 708*x^12 - 9630*x^11 + 133026*x^10 - 824449*x^9 + 4611324*x^8 - 10911485*x^7 - 723492*x^6 + 271409820*x^5 - 1328274288*x^4 + 3021383088*x^3 + 24795072*x^2 - 36616748544*x + 88119373824);
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
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
| 3.1.220323.1, 5.1.48542224329.1 |
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 | ${\href{/padicField/2.2.0.1}{2} }^{7}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | ${\href{/padicField/5.2.0.1}{2} }^{7}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $15$ | ${\href{/padicField/11.2.0.1}{2} }^{7}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.3.0.1}{3} }^{5}$ | ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $15$ | ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.3.0.1}{3} }^{5}$ | ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(271\)
| Deg $15$ | $15$ | $1$ | $14$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 2.220323.3t2.a.a | $2$ | $ 3 \cdot 271^{2}$ | 3.1.220323.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.220323.5t2.a.b | $2$ | $ 3 \cdot 271^{2}$ | 5.1.48542224329.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.220323.5t2.a.a | $2$ | $ 3 \cdot 271^{2}$ | 5.1.48542224329.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.220323.15t2.a.d | $2$ | $ 3 \cdot 271^{2}$ | 15.1.25201062723784606393449803925478522347.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.220323.15t2.a.b | $2$ | $ 3 \cdot 271^{2}$ | 15.1.25201062723784606393449803925478522347.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.220323.15t2.a.c | $2$ | $ 3 \cdot 271^{2}$ | 15.1.25201062723784606393449803925478522347.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.220323.15t2.a.a | $2$ | $ 3 \cdot 271^{2}$ | 15.1.25201062723784606393449803925478522347.1 | $D_{15}$ (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 *.