Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 80*x^13 + 460*x^12 + 2265*x^11 - 15553*x^10 - 24090*x^9 + 392310*x^8 - 976120*x^7 + 1638320*x^6 - 2453272*x^5 + 68720*x^4 - 5194480*x^3 + 231378160*x^2 + 420777040*x - 520184176)
gp: K = bnfinit(y^15 - 5*y^14 - 80*y^13 + 460*y^12 + 2265*y^11 - 15553*y^10 - 24090*y^9 + 392310*y^8 - 976120*y^7 + 1638320*y^6 - 2453272*y^5 + 68720*y^4 - 5194480*y^3 + 231378160*y^2 + 420777040*y - 520184176, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 5*x^14 - 80*x^13 + 460*x^12 + 2265*x^11 - 15553*x^10 - 24090*x^9 + 392310*x^8 - 976120*x^7 + 1638320*x^6 - 2453272*x^5 + 68720*x^4 - 5194480*x^3 + 231378160*x^2 + 420777040*x - 520184176);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 5*x^14 - 80*x^13 + 460*x^12 + 2265*x^11 - 15553*x^10 - 24090*x^9 + 392310*x^8 - 976120*x^7 + 1638320*x^6 - 2453272*x^5 + 68720*x^4 - 5194480*x^3 + 231378160*x^2 + 420777040*x - 520184176)
\( x^{15} - 5 x^{14} - 80 x^{13} + 460 x^{12} + 2265 x^{11} - 15553 x^{10} - 24090 x^{9} + 392310 x^{8} + \cdots - 520184176 \)
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: | $[3, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(140685733870773315116885561600000000000000000\)
\(\medspace = 2^{23}\cdot 5^{17}\cdot 89^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(877.44\) | 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: | $2^{19/10}5^{71/60}89^{9/10}\approx 1424.0410311070943$ | ||
| Ramified primes: |
\(2\), \(5\), \(89\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{890}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}-\frac{1}{5}$, $\frac{1}{10}a^{6}-\frac{1}{10}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{10}a^{7}-\frac{1}{2}a^{3}+\frac{2}{5}a^{2}$, $\frac{1}{10}a^{8}-\frac{1}{2}a^{4}+\frac{2}{5}a^{3}$, $\frac{1}{50}a^{9}+\frac{1}{50}a^{8}+\frac{1}{50}a^{7}+\frac{1}{50}a^{6}-\frac{2}{25}a^{5}+\frac{2}{25}a^{4}+\frac{2}{25}a^{3}+\frac{2}{25}a^{2}+\frac{2}{25}a-\frac{8}{25}$, $\frac{1}{8900}a^{10}-\frac{13}{8900}a^{9}+\frac{53}{2225}a^{8}+\frac{63}{2225}a^{7}+\frac{287}{8900}a^{6}-\frac{107}{1780}a^{5}-\frac{888}{2225}a^{4}-\frac{588}{2225}a^{3}-\frac{898}{2225}a^{2}-\frac{738}{2225}a+\frac{161}{2225}$, $\frac{1}{8900}a^{11}+\frac{43}{8900}a^{9}+\frac{169}{4450}a^{8}+\frac{3}{8900}a^{7}-\frac{91}{2225}a^{6}+\frac{173}{8900}a^{5}+\frac{211}{4450}a^{4}-\frac{87}{2225}a^{3}-\frac{397}{2225}a^{2}+\frac{357}{2225}a-\frac{577}{2225}$, $\frac{1}{44500}a^{12}-\frac{1}{22250}a^{11}+\frac{1}{44500}a^{10}-\frac{27}{4450}a^{9}-\frac{349}{8900}a^{8}+\frac{219}{22250}a^{7}+\frac{239}{44500}a^{6}-\frac{831}{22250}a^{5}-\frac{379}{2225}a^{4}+\frac{676}{2225}a^{3}+\frac{1309}{11125}a^{2}+\frac{5497}{11125}a+\frac{1334}{11125}$, $\frac{1}{89000}a^{13}-\frac{1}{89000}a^{12}+\frac{1}{22250}a^{11}-\frac{1}{22250}a^{10}-\frac{159}{17800}a^{9}+\frac{3163}{89000}a^{8}+\frac{361}{44500}a^{7}-\frac{1419}{44500}a^{6}-\frac{969}{11125}a^{5}-\frac{159}{4450}a^{4}+\frac{6409}{22250}a^{3}+\frac{238}{11125}a^{2}+\frac{2198}{11125}a+\frac{2757}{11125}$, $\frac{1}{90\!\cdots\!00}a^{14}+\frac{49\!\cdots\!41}{90\!\cdots\!00}a^{13}+\frac{91\!\cdots\!49}{11\!\cdots\!25}a^{12}+\frac{40\!\cdots\!37}{45\!\cdots\!00}a^{11}-\frac{51\!\cdots\!47}{10\!\cdots\!00}a^{10}-\frac{67\!\cdots\!27}{90\!\cdots\!00}a^{9}-\frac{57\!\cdots\!91}{45\!\cdots\!00}a^{8}-\frac{36\!\cdots\!03}{11\!\cdots\!25}a^{7}-\frac{42\!\cdots\!21}{11\!\cdots\!25}a^{6}+\frac{87\!\cdots\!19}{22\!\cdots\!50}a^{5}+\frac{61\!\cdots\!19}{22\!\cdots\!50}a^{4}+\frac{49\!\cdots\!77}{11\!\cdots\!25}a^{3}+\frac{80\!\cdots\!04}{11\!\cdots\!25}a^{2}+\frac{17\!\cdots\!93}{11\!\cdots\!25}a+\frac{54\!\cdots\!04}{11\!\cdots\!25}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{4}\times C_{40}$, which has order $160$ (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 $2$)
| 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: |
$\frac{26\!\cdots\!48}{50\!\cdots\!45}a^{14}-\frac{58\!\cdots\!32}{25\!\cdots\!25}a^{13}-\frac{10\!\cdots\!32}{25\!\cdots\!25}a^{12}+\frac{20\!\cdots\!72}{10\!\cdots\!49}a^{11}+\frac{27\!\cdots\!44}{25\!\cdots\!25}a^{10}-\frac{31\!\cdots\!12}{50\!\cdots\!45}a^{9}-\frac{30\!\cdots\!36}{25\!\cdots\!25}a^{8}+\frac{40\!\cdots\!64}{25\!\cdots\!25}a^{7}-\frac{39\!\cdots\!04}{10\!\cdots\!49}a^{6}+\frac{38\!\cdots\!12}{25\!\cdots\!25}a^{5}-\frac{29\!\cdots\!56}{50\!\cdots\!45}a^{4}+\frac{29\!\cdots\!68}{25\!\cdots\!25}a^{3}-\frac{26\!\cdots\!32}{25\!\cdots\!25}a^{2}+\frac{10\!\cdots\!44}{10\!\cdots\!49}a-\frac{42\!\cdots\!31}{25\!\cdots\!25}$, $\frac{89\!\cdots\!51}{40\!\cdots\!96}a^{14}-\frac{48\!\cdots\!86}{50\!\cdots\!45}a^{13}-\frac{17\!\cdots\!09}{10\!\cdots\!90}a^{12}+\frac{42\!\cdots\!39}{50\!\cdots\!45}a^{11}+\frac{48\!\cdots\!91}{10\!\cdots\!90}a^{10}-\frac{26\!\cdots\!89}{10\!\cdots\!49}a^{9}-\frac{27\!\cdots\!23}{50\!\cdots\!45}a^{8}+\frac{35\!\cdots\!64}{50\!\cdots\!45}a^{7}-\frac{32\!\cdots\!17}{20\!\cdots\!80}a^{6}+\frac{28\!\cdots\!49}{50\!\cdots\!45}a^{5}-\frac{40\!\cdots\!95}{20\!\cdots\!98}a^{4}+\frac{19\!\cdots\!84}{50\!\cdots\!45}a^{3}-\frac{17\!\cdots\!47}{50\!\cdots\!45}a^{2}+\frac{23\!\cdots\!59}{50\!\cdots\!45}a-\frac{19\!\cdots\!27}{50\!\cdots\!45}$, $\frac{17\!\cdots\!93}{50\!\cdots\!50}a^{14}+\frac{80\!\cdots\!01}{10\!\cdots\!90}a^{13}-\frac{10\!\cdots\!53}{25\!\cdots\!25}a^{12}-\frac{19\!\cdots\!66}{25\!\cdots\!25}a^{11}+\frac{10\!\cdots\!97}{50\!\cdots\!50}a^{10}+\frac{17\!\cdots\!49}{50\!\cdots\!50}a^{9}-\frac{51\!\cdots\!59}{10\!\cdots\!90}a^{8}-\frac{99\!\cdots\!69}{25\!\cdots\!25}a^{7}+\frac{20\!\cdots\!57}{25\!\cdots\!25}a^{6}+\frac{85\!\cdots\!38}{25\!\cdots\!25}a^{5}-\frac{24\!\cdots\!67}{50\!\cdots\!50}a^{4}-\frac{15\!\cdots\!56}{50\!\cdots\!45}a^{3}+\frac{10\!\cdots\!72}{25\!\cdots\!25}a^{2}+\frac{11\!\cdots\!84}{25\!\cdots\!25}a-\frac{13\!\cdots\!49}{25\!\cdots\!25}$, $\frac{17\!\cdots\!03}{90\!\cdots\!00}a^{14}-\frac{59\!\cdots\!01}{45\!\cdots\!50}a^{13}-\frac{28\!\cdots\!91}{22\!\cdots\!25}a^{12}+\frac{10\!\cdots\!51}{90\!\cdots\!00}a^{11}+\frac{20\!\cdots\!77}{10\!\cdots\!00}a^{10}-\frac{15\!\cdots\!53}{45\!\cdots\!50}a^{9}+\frac{94\!\cdots\!17}{45\!\cdots\!50}a^{8}+\frac{62\!\cdots\!43}{90\!\cdots\!00}a^{7}-\frac{14\!\cdots\!51}{45\!\cdots\!50}a^{6}+\frac{43\!\cdots\!47}{45\!\cdots\!50}a^{5}-\frac{53\!\cdots\!93}{22\!\cdots\!25}a^{4}+\frac{10\!\cdots\!17}{22\!\cdots\!25}a^{3}-\frac{23\!\cdots\!51}{22\!\cdots\!25}a^{2}+\frac{14\!\cdots\!94}{22\!\cdots\!25}a-\frac{11\!\cdots\!43}{22\!\cdots\!25}$, $\frac{71\!\cdots\!51}{36\!\cdots\!40}a^{14}+\frac{17\!\cdots\!31}{45\!\cdots\!50}a^{13}-\frac{30\!\cdots\!91}{18\!\cdots\!00}a^{12}-\frac{54\!\cdots\!01}{90\!\cdots\!00}a^{11}+\frac{88\!\cdots\!17}{20\!\cdots\!00}a^{10}-\frac{28\!\cdots\!19}{45\!\cdots\!05}a^{9}-\frac{16\!\cdots\!33}{18\!\cdots\!00}a^{8}+\frac{23\!\cdots\!21}{90\!\cdots\!00}a^{7}-\frac{20\!\cdots\!29}{45\!\cdots\!50}a^{6}+\frac{26\!\cdots\!21}{45\!\cdots\!50}a^{5}-\frac{80\!\cdots\!43}{45\!\cdots\!05}a^{4}-\frac{57\!\cdots\!79}{45\!\cdots\!50}a^{3}-\frac{21\!\cdots\!72}{22\!\cdots\!25}a^{2}-\frac{25\!\cdots\!29}{22\!\cdots\!25}a+\frac{39\!\cdots\!91}{22\!\cdots\!25}$, $\frac{47\!\cdots\!13}{11\!\cdots\!00}a^{14}-\frac{14\!\cdots\!53}{50\!\cdots\!45}a^{13}-\frac{45\!\cdots\!79}{10\!\cdots\!00}a^{12}+\frac{15\!\cdots\!54}{50\!\cdots\!45}a^{11}+\frac{20\!\cdots\!83}{10\!\cdots\!00}a^{10}-\frac{18\!\cdots\!71}{25\!\cdots\!25}a^{9}-\frac{10\!\cdots\!17}{20\!\cdots\!80}a^{8}+\frac{13\!\cdots\!87}{25\!\cdots\!25}a^{7}+\frac{60\!\cdots\!28}{10\!\cdots\!49}a^{6}-\frac{10\!\cdots\!44}{25\!\cdots\!25}a^{5}-\frac{10\!\cdots\!62}{25\!\cdots\!25}a^{4}+\frac{53\!\cdots\!56}{50\!\cdots\!45}a^{3}+\frac{80\!\cdots\!14}{25\!\cdots\!25}a^{2}+\frac{10\!\cdots\!56}{50\!\cdots\!45}a-\frac{10\!\cdots\!33}{25\!\cdots\!25}$, $\frac{10\!\cdots\!79}{90\!\cdots\!00}a^{14}-\frac{31\!\cdots\!97}{45\!\cdots\!00}a^{13}-\frac{16\!\cdots\!43}{18\!\cdots\!00}a^{12}+\frac{15\!\cdots\!69}{22\!\cdots\!50}a^{11}+\frac{29\!\cdots\!91}{10\!\cdots\!00}a^{10}-\frac{30\!\cdots\!36}{11\!\cdots\!25}a^{9}-\frac{33\!\cdots\!97}{90\!\cdots\!00}a^{8}+\frac{32\!\cdots\!03}{45\!\cdots\!50}a^{7}-\frac{51\!\cdots\!81}{45\!\cdots\!00}a^{6}-\frac{16\!\cdots\!39}{45\!\cdots\!00}a^{5}+\frac{72\!\cdots\!21}{22\!\cdots\!50}a^{4}+\frac{95\!\cdots\!29}{22\!\cdots\!50}a^{3}-\frac{36\!\cdots\!33}{22\!\cdots\!25}a^{2}-\frac{15\!\cdots\!33}{11\!\cdots\!25}a+\frac{11\!\cdots\!38}{11\!\cdots\!25}$, $\frac{36\!\cdots\!13}{45\!\cdots\!00}a^{14}+\frac{28\!\cdots\!99}{90\!\cdots\!00}a^{13}-\frac{61\!\cdots\!97}{90\!\cdots\!00}a^{12}-\frac{29\!\cdots\!01}{45\!\cdots\!00}a^{11}+\frac{11\!\cdots\!53}{50\!\cdots\!00}a^{10}+\frac{19\!\cdots\!53}{90\!\cdots\!00}a^{9}-\frac{27\!\cdots\!13}{90\!\cdots\!00}a^{8}+\frac{14\!\cdots\!93}{11\!\cdots\!25}a^{7}+\frac{13\!\cdots\!27}{45\!\cdots\!00}a^{6}+\frac{45\!\cdots\!71}{45\!\cdots\!00}a^{5}+\frac{46\!\cdots\!49}{22\!\cdots\!50}a^{4}+\frac{23\!\cdots\!33}{11\!\cdots\!25}a^{3}-\frac{29\!\cdots\!84}{11\!\cdots\!25}a^{2}-\frac{49\!\cdots\!44}{11\!\cdots\!25}a+\frac{60\!\cdots\!78}{11\!\cdots\!25}$
| 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: | \( 308642886210973.75 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 308642886210973.75 \cdot 160}{2\cdot\sqrt{140685733870773315116885561600000000000000000}}\cr\approx \mathstrut & 1.02468511021684 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 80*x^13 + 460*x^12 + 2265*x^11 - 15553*x^10 - 24090*x^9 + 392310*x^8 - 976120*x^7 + 1638320*x^6 - 2453272*x^5 + 68720*x^4 - 5194480*x^3 + 231378160*x^2 + 420777040*x - 520184176)
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 - 5*x^14 - 80*x^13 + 460*x^12 + 2265*x^11 - 15553*x^10 - 24090*x^9 + 392310*x^8 - 976120*x^7 + 1638320*x^6 - 2453272*x^5 + 68720*x^4 - 5194480*x^3 + 231378160*x^2 + 420777040*x - 520184176, 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 - 5*x^14 - 80*x^13 + 460*x^12 + 2265*x^11 - 15553*x^10 - 24090*x^9 + 392310*x^8 - 976120*x^7 + 1638320*x^6 - 2453272*x^5 + 68720*x^4 - 5194480*x^3 + 231378160*x^2 + 420777040*x - 520184176);
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 - 5*x^14 - 80*x^13 + 460*x^12 + 2265*x^11 - 15553*x^10 - 24090*x^9 + 392310*x^8 - 976120*x^7 + 1638320*x^6 - 2453272*x^5 + 68720*x^4 - 5194480*x^3 + 231378160*x^2 + 420777040*x - 520184176);
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
$S_3\times F_5$ (as 15T11):
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 120 |
| The 15 conjugacy class representatives for $F_5 \times S_3$ |
| Character table for $F_5 \times S_3$ |
Intermediate fields
| 3.3.17800.1, 5.1.3137112050000.2 |
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 30 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 | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | R | ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.5.4.1 | $x^{5} + 2$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ |
| 2.10.19.1 | $x^{10} + 4 x^{5} + 2$ | $10$ | $1$ | $19$ | $F_{5}\times C_2$ | $[3]_{5}^{4}$ | |
|
\(5\)
| 5.15.17.3 | $x^{15} + 5 x^{3} + 5$ | $15$ | $1$ | $17$ | $F_5 \times S_3$ | $[5/4]_{12}^{2}$ |
|
\(89\)
| 89.5.4.1 | $x^{5} + 89$ | $5$ | $1$ | $4$ | $D_{5}$ | $[\ ]_{5}^{2}$ |
| 89.10.9.1 | $x^{10} + 89$ | $10$ | $1$ | $9$ | $D_{10}$ | $[\ ]_{10}^{2}$ |