Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 126*x^13 - 1008*x^12 - 47968*x^11 - 737248*x^10 - 5886360*x^9 - 48520512*x^8 - 378179296*x^7 - 1954203776*x^6 - 6255936640*x^5 - 12603064320*x^4 - 16068774400*x^3 - 12611020800*x^2 - 5569536000*x - 1060864000)
gp: K = bnfinit(y^15 - 126*y^13 - 1008*y^12 - 47968*y^11 - 737248*y^10 - 5886360*y^9 - 48520512*y^8 - 378179296*y^7 - 1954203776*y^6 - 6255936640*y^5 - 12603064320*y^4 - 16068774400*y^3 - 12611020800*y^2 - 5569536000*y - 1060864000, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 126*x^13 - 1008*x^12 - 47968*x^11 - 737248*x^10 - 5886360*x^9 - 48520512*x^8 - 378179296*x^7 - 1954203776*x^6 - 6255936640*x^5 - 12603064320*x^4 - 16068774400*x^3 - 12611020800*x^2 - 5569536000*x - 1060864000);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 126*x^13 - 1008*x^12 - 47968*x^11 - 737248*x^10 - 5886360*x^9 - 48520512*x^8 - 378179296*x^7 - 1954203776*x^6 - 6255936640*x^5 - 12603064320*x^4 - 16068774400*x^3 - 12611020800*x^2 - 5569536000*x - 1060864000)
\( x^{15} - 126 x^{13} - 1008 x^{12} - 47968 x^{11} - 737248 x^{10} - 5886360 x^{9} - 48520512 x^{8} + \cdots - 1060864000 \)
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: | $[11, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(6242427558646320395593447809832981520060416000000\)
\(\medspace = 2^{18}\cdot 5^{6}\cdot 7^{4}\cdot 19^{10}\cdot 37^{4}\cdot 1481^{2}\cdot 158699^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(1790.70\) | 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: | $2^{9/4}5^{1/2}7^{4/5}19^{2/3}37^{4/5}1481^{1/2}158699^{1/2}\approx 98971312.27079217$ | ||
| Ramified primes: |
\(2\), \(5\), \(7\), \(19\), \(37\), \(1481\), \(158699\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{20}a^{5}-\frac{1}{10}a^{3}+\frac{1}{5}a^{2}$, $\frac{1}{40}a^{6}-\frac{1}{20}a^{4}+\frac{1}{10}a^{3}$, $\frac{1}{80}a^{7}-\frac{1}{40}a^{5}+\frac{1}{20}a^{4}-\frac{1}{2}a$, $\frac{1}{800}a^{8}+\frac{3}{400}a^{6}+\frac{1}{100}a^{5}+\frac{2}{25}a^{4}-\frac{1}{50}a^{3}+\frac{7}{100}a^{2}-\frac{1}{5}$, $\frac{1}{3200}a^{9}-\frac{7}{1600}a^{7}-\frac{1}{100}a^{6}-\frac{1}{200}a^{5}+\frac{3}{25}a^{4}+\frac{17}{400}a^{3}+\frac{1}{10}a^{2}+\frac{9}{20}a$, $\frac{1}{6400}a^{10}+\frac{1}{3200}a^{8}-\frac{1}{200}a^{7}-\frac{1}{80}a^{6}-\frac{1}{50}a^{5}-\frac{3}{160}a^{4}+\frac{11}{100}a^{3}-\frac{7}{200}a^{2}-\frac{2}{5}$, $\frac{1}{128000}a^{11}+\frac{7}{64000}a^{9}-\frac{3}{8000}a^{8}+\frac{89}{16000}a^{7}-\frac{9}{4000}a^{6}-\frac{27}{3200}a^{5}-\frac{83}{2000}a^{4}+\frac{153}{1000}a^{3}-\frac{87}{1000}a^{2}-\frac{97}{200}a+\frac{1}{50}$, $\frac{1}{1280000}a^{12}+\frac{47}{640000}a^{10}+\frac{3}{20000}a^{9}-\frac{11}{160000}a^{8}-\frac{97}{20000}a^{7}+\frac{277}{32000}a^{6}-\frac{323}{20000}a^{5}-\frac{341}{5000}a^{4}-\frac{117}{625}a^{3}-\frac{249}{2000}a^{2}+\frac{34}{125}a+\frac{9}{25}$, $\frac{1}{5120000000}a^{13}-\frac{31}{128000000}a^{12}+\frac{5817}{2560000000}a^{11}+\frac{451}{160000000}a^{10}-\frac{2993}{20000000}a^{9}-\frac{36319}{160000000}a^{8}-\frac{148343}{128000000}a^{7}-\frac{75207}{20000000}a^{6}+\frac{2239237}{160000000}a^{5}+\frac{717073}{40000000}a^{4}-\frac{1208209}{8000000}a^{3}-\frac{130799}{1000000}a^{2}+\frac{181959}{400000}a+\frac{31017}{100000}$, $\frac{1}{655360000000}a^{14}-\frac{11}{163840000000}a^{13}+\frac{60297}{327680000000}a^{12}-\frac{258913}{81920000000}a^{11}-\frac{63003}{1280000000}a^{10}+\frac{2478657}{20480000000}a^{9}-\frac{32003811}{81920000000}a^{8}+\frac{30777559}{20480000000}a^{7}-\frac{198911339}{20480000000}a^{6}+\frac{10367617}{640000000}a^{5}+\frac{39834263}{5120000000}a^{4}-\frac{1347289}{256000000}a^{3}-\frac{33023413}{256000000}a^{2}-\frac{1598821}{6400000}a-\frac{1323917}{3200000}$
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
Trivial group, which has order $1$ (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: | $12$ | 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: |
$\frac{252379683}{655360000000}a^{14}-\frac{69969281}{163840000000}a^{13}-\frac{15745661509}{327680000000}a^{12}-\frac{5486775207}{16384000000}a^{11}-\frac{23169030641}{1280000000}a^{10}-\frac{5403443670301}{20480000000}a^{9}-\frac{161720436492041}{81920000000}a^{8}-\frac{337804657853083}{20480000000}a^{7}-\frac{521560939566061}{4096000000}a^{6}-\frac{391204535797341}{640000000}a^{5}-\frac{88\!\cdots\!27}{5120000000}a^{4}-\frac{30015474900579}{10240000}a^{3}-\frac{750187168537023}{256000000}a^{2}-\frac{10226971451599}{6400000}a-\frac{1174521975367}{3200000}$, $\frac{252379683}{655360000000}a^{14}-\frac{69969281}{163840000000}a^{13}-\frac{15745661509}{327680000000}a^{12}-\frac{5486775207}{16384000000}a^{11}-\frac{23169030641}{1280000000}a^{10}-\frac{5403443670301}{20480000000}a^{9}-\frac{161720436492041}{81920000000}a^{8}-\frac{337804657853083}{20480000000}a^{7}-\frac{521560939566061}{4096000000}a^{6}-\frac{391204535797341}{640000000}a^{5}-\frac{88\!\cdots\!27}{5120000000}a^{4}-\frac{30015474900579}{10240000}a^{3}-\frac{750187168537023}{256000000}a^{2}-\frac{10226971451599}{6400000}a-\frac{1174525175367}{3200000}$, $\frac{32\!\cdots\!73}{655360000000}a^{14}-\frac{12\!\cdots\!99}{163840000000}a^{13}-\frac{19\!\cdots\!39}{327680000000}a^{12}-\frac{33\!\cdots\!81}{81920000000}a^{11}-\frac{29\!\cdots\!43}{1280000000}a^{10}-\frac{67\!\cdots\!43}{20480000000}a^{9}-\frac{19\!\cdots\!19}{81920000000}a^{8}-\frac{41\!\cdots\!53}{20480000000}a^{7}-\frac{31\!\cdots\!43}{20480000000}a^{6}-\frac{46\!\cdots\!03}{640000000}a^{5}-\frac{10\!\cdots\!33}{5120000000}a^{4}-\frac{81\!\cdots\!33}{256000000}a^{3}-\frac{77\!\cdots\!17}{256000000}a^{2}-\frac{20\!\cdots\!13}{1280000}a-\frac{10\!\cdots\!73}{3200000}$, $\frac{8683901348901}{10240000000}a^{14}+\frac{25143934147461}{5120000000}a^{13}+\frac{68524826276677}{5120000000}a^{12}+\frac{498948951865597}{512000000}a^{11}-\frac{16\!\cdots\!91}{160000000}a^{10}-\frac{80\!\cdots\!47}{320000000}a^{9}-\frac{22\!\cdots\!27}{1280000000}a^{8}-\frac{10\!\cdots\!27}{640000000}a^{7}-\frac{94\!\cdots\!47}{64000000}a^{6}-\frac{11\!\cdots\!57}{160000000}a^{5}-\frac{16\!\cdots\!19}{80000000}a^{4}-\frac{11\!\cdots\!21}{320000}a^{3}-\frac{13\!\cdots\!81}{4000000}a^{2}-\frac{69\!\cdots\!87}{400000}a-\frac{38\!\cdots\!23}{100000}$, $\frac{44\!\cdots\!51}{327680000000}a^{14}-\frac{12\!\cdots\!69}{81920000000}a^{13}-\frac{27\!\cdots\!13}{163840000000}a^{12}-\frac{48\!\cdots\!99}{40960000000}a^{11}-\frac{81\!\cdots\!81}{128000000}a^{10}-\frac{18\!\cdots\!77}{2048000000}a^{9}-\frac{28\!\cdots\!29}{40960000000}a^{8}-\frac{59\!\cdots\!71}{10240000000}a^{7}-\frac{45\!\cdots\!97}{10240000000}a^{6}-\frac{13\!\cdots\!49}{64000000}a^{5}-\frac{15\!\cdots\!23}{2560000000}a^{4}-\frac{12\!\cdots\!67}{128000000}a^{3}-\frac{12\!\cdots\!27}{128000000}a^{2}-\frac{17\!\cdots\!07}{3200000}a-\frac{19\!\cdots\!03}{1600000}$, $\frac{27076961}{655360000000}a^{14}-\frac{264888267}{163840000000}a^{13}-\frac{5656916503}{327680000000}a^{12}+\frac{626949239}{3276800000}a^{11}+\frac{2670214893}{1280000000}a^{10}+\frac{1216429997473}{20480000000}a^{9}+\frac{110249353654013}{81920000000}a^{8}+\frac{268434788426839}{20480000000}a^{7}+\frac{55683143530941}{819200000}a^{6}+\frac{134783279733693}{640000000}a^{5}+\frac{21\!\cdots\!11}{5120000000}a^{4}+\frac{26092668615547}{51200000}a^{3}+\frac{100224581368939}{256000000}a^{2}+\frac{1088015515787}{6400000}a+\frac{102230066931}{3200000}$, $\frac{31\!\cdots\!71}{1310720000}a^{14}-\frac{26\!\cdots\!17}{8192000000}a^{13}-\frac{96\!\cdots\!13}{3276800000}a^{12}-\frac{81\!\cdots\!39}{4096000000}a^{11}-\frac{70\!\cdots\!73}{64000000}a^{10}-\frac{16\!\cdots\!33}{1024000000}a^{9}-\frac{48\!\cdots\!57}{4096000000}a^{8}-\frac{20\!\cdots\!59}{204800000}a^{7}-\frac{77\!\cdots\!17}{1024000000}a^{6}-\frac{11\!\cdots\!13}{32000000}a^{5}-\frac{25\!\cdots\!39}{256000000}a^{4}-\frac{20\!\cdots\!67}{12800000}a^{3}-\frac{19\!\cdots\!11}{12800000}a^{2}-\frac{26\!\cdots\!59}{320000}a-\frac{28\!\cdots\!39}{160000}$, $\frac{18\!\cdots\!07}{40960000000}a^{14}-\frac{56\!\cdots\!11}{10240000000}a^{13}-\frac{11\!\cdots\!01}{20480000000}a^{12}-\frac{19\!\cdots\!69}{5120000000}a^{11}-\frac{33\!\cdots\!09}{160000000}a^{10}-\frac{38\!\cdots\!17}{1280000000}a^{9}-\frac{11\!\cdots\!41}{5120000000}a^{8}-\frac{23\!\cdots\!77}{1280000000}a^{7}-\frac{18\!\cdots\!57}{1280000000}a^{6}-\frac{10\!\cdots\!53}{160000000}a^{5}-\frac{60\!\cdots\!87}{320000000}a^{4}-\frac{50\!\cdots\!17}{16000000}a^{3}-\frac{49\!\cdots\!63}{16000000}a^{2}-\frac{13\!\cdots\!33}{8000}a-\frac{74\!\cdots\!97}{200000}$, $\frac{63\!\cdots\!27}{65536000000}a^{14}-\frac{12\!\cdots\!69}{81920000000}a^{13}-\frac{39\!\cdots\!17}{32768000000}a^{12}-\frac{32\!\cdots\!83}{40960000000}a^{11}-\frac{29\!\cdots\!01}{640000000}a^{10}-\frac{66\!\cdots\!21}{10240000000}a^{9}-\frac{19\!\cdots\!49}{40960000000}a^{8}-\frac{16\!\cdots\!19}{409600000}a^{7}-\frac{31\!\cdots\!49}{10240000000}a^{6}-\frac{45\!\cdots\!31}{320000000}a^{5}-\frac{99\!\cdots\!23}{2560000000}a^{4}-\frac{80\!\cdots\!59}{128000000}a^{3}-\frac{76\!\cdots\!27}{128000000}a^{2}-\frac{98\!\cdots\!63}{3200000}a-\frac{10\!\cdots\!23}{1600000}$, $\frac{39\!\cdots\!87}{655360000000}a^{14}-\frac{12\!\cdots\!69}{163840000000}a^{13}-\frac{24\!\cdots\!01}{327680000000}a^{12}-\frac{84\!\cdots\!07}{16384000000}a^{11}-\frac{36\!\cdots\!89}{1280000000}a^{10}-\frac{84\!\cdots\!29}{20480000000}a^{9}-\frac{24\!\cdots\!09}{81920000000}a^{8}-\frac{52\!\cdots\!87}{20480000000}a^{7}-\frac{80\!\cdots\!81}{4096000000}a^{6}-\frac{59\!\cdots\!89}{640000000}a^{5}-\frac{13\!\cdots\!23}{5120000000}a^{4}-\frac{22\!\cdots\!67}{51200000}a^{3}-\frac{10\!\cdots\!27}{256000000}a^{2}-\frac{14\!\cdots\!31}{6400000}a-\frac{16\!\cdots\!83}{3200000}$, $\frac{77\!\cdots\!27}{131072000000}a^{14}-\frac{14\!\cdots\!89}{163840000000}a^{13}-\frac{47\!\cdots\!97}{65536000000}a^{12}-\frac{40\!\cdots\!23}{81920000000}a^{11}-\frac{35\!\cdots\!81}{1280000000}a^{10}-\frac{81\!\cdots\!01}{20480000000}a^{9}-\frac{23\!\cdots\!69}{81920000000}a^{8}-\frac{20\!\cdots\!07}{819200000}a^{7}-\frac{38\!\cdots\!69}{20480000000}a^{6}-\frac{57\!\cdots\!61}{640000000}a^{5}-\frac{12\!\cdots\!63}{5120000000}a^{4}-\frac{10\!\cdots\!79}{256000000}a^{3}-\frac{10\!\cdots\!87}{256000000}a^{2}-\frac{13\!\cdots\!03}{6400000}a-\frac{15\!\cdots\!63}{3200000}$, $\frac{17\!\cdots\!29}{655360000000}a^{14}-\frac{18\!\cdots\!91}{163840000000}a^{13}-\frac{92\!\cdots\!27}{327680000000}a^{12}-\frac{11\!\cdots\!01}{81920000000}a^{11}-\frac{30\!\cdots\!51}{256000000}a^{10}-\frac{93\!\cdots\!27}{6553600}a^{9}-\frac{76\!\cdots\!31}{81920000000}a^{8}-\frac{17\!\cdots\!09}{20480000000}a^{7}-\frac{12\!\cdots\!03}{20480000000}a^{6}-\frac{31\!\cdots\!83}{128000000}a^{5}-\frac{29\!\cdots\!97}{5120000000}a^{4}-\frac{21\!\cdots\!33}{256000000}a^{3}-\frac{17\!\cdots\!53}{256000000}a^{2}-\frac{20\!\cdots\!33}{6400000}a-\frac{20\!\cdots\!17}{3200000}$
| 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: | \( 33525688191400000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{2}\cdot 33525688191400000000 \cdot 1}{2\cdot\sqrt{6242427558646320395593447809832981520060416000000}}\cr\approx \mathstrut & 0.542451156261110 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 126*x^13 - 1008*x^12 - 47968*x^11 - 737248*x^10 - 5886360*x^9 - 48520512*x^8 - 378179296*x^7 - 1954203776*x^6 - 6255936640*x^5 - 12603064320*x^4 - 16068774400*x^3 - 12611020800*x^2 - 5569536000*x - 1060864000)
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 - 126*x^13 - 1008*x^12 - 47968*x^11 - 737248*x^10 - 5886360*x^9 - 48520512*x^8 - 378179296*x^7 - 1954203776*x^6 - 6255936640*x^5 - 12603064320*x^4 - 16068774400*x^3 - 12611020800*x^2 - 5569536000*x - 1060864000, 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 - 126*x^13 - 1008*x^12 - 47968*x^11 - 737248*x^10 - 5886360*x^9 - 48520512*x^8 - 378179296*x^7 - 1954203776*x^6 - 6255936640*x^5 - 12603064320*x^4 - 16068774400*x^3 - 12611020800*x^2 - 5569536000*x - 1060864000);
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 - 126*x^13 - 1008*x^12 - 47968*x^11 - 737248*x^10 - 5886360*x^9 - 48520512*x^8 - 378179296*x^7 - 1954203776*x^6 - 6255936640*x^5 - 12603064320*x^4 - 16068774400*x^3 - 12611020800*x^2 - 5569536000*x - 1060864000);
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
$A_5^3.A_4$ (as 15T98):
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 non-solvable group of order 2592000 |
| The 71 conjugacy class representatives for $A_5^3.A_4$ |
| Character table for $A_5^3.A_4$ |
Intermediate fields
| 3.3.361.1 |
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 18 sibling: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Degree 45 sibling: | 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 | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }$ | R | R | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | R | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | $15$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | $15$ | $15$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.6.9.8 | $x^{6} - 8 x^{5} + 62 x^{4} + 224 x^{3} + 316 x^{2} + 1184 x + 1608$ | $2$ | $3$ | $9$ | $A_4\times C_2$ | $[2, 2, 3]^{3}$ | |
| 2.6.9.8 | $x^{6} - 8 x^{5} + 62 x^{4} + 224 x^{3} + 316 x^{2} + 1184 x + 1608$ | $2$ | $3$ | $9$ | $A_4\times C_2$ | $[2, 2, 3]^{3}$ | |
|
\(5\)
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.5.4.1 | $x^{5} + 7$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
|
\(19\)
| 19.15.10.1 | $x^{15} + 95 x^{12} + 15 x^{11} + 51 x^{10} + 3610 x^{9} - 4275 x^{8} - 28995 x^{7} + 69100 x^{6} - 31623 x^{5} + 835620 x^{4} + 700180 x^{3} + 1194855 x^{2} - 1636410 x + 2391332$ | $3$ | $5$ | $10$ | $C_{15}$ | $[\ ]_{3}^{5}$ |
|
\(37\)
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.5.4.1 | $x^{5} + 37$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 37.5.0.1 | $x^{5} + 10 x + 35$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
|
\(1481\)
| $\Q_{1481}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1481}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1481}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1481}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1481}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
|
\(158699\)
| $\Q_{158699}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{158699}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |