Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 8*x^13 + 43*x^12 - 98*x^11 + 206*x^10 - 138*x^9 - 450*x^8 + 935*x^7 - 737*x^6 - 670*x^5 + 1953*x^4 + 601*x^3 - 913*x^2 - 428*x - 43)
gp: K = bnfinit(y^15 - 3*y^14 - 8*y^13 + 43*y^12 - 98*y^11 + 206*y^10 - 138*y^9 - 450*y^8 + 935*y^7 - 737*y^6 - 670*y^5 + 1953*y^4 + 601*y^3 - 913*y^2 - 428*y - 43, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 3*x^14 - 8*x^13 + 43*x^12 - 98*x^11 + 206*x^10 - 138*x^9 - 450*x^8 + 935*x^7 - 737*x^6 - 670*x^5 + 1953*x^4 + 601*x^3 - 913*x^2 - 428*x - 43);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 3*x^14 - 8*x^13 + 43*x^12 - 98*x^11 + 206*x^10 - 138*x^9 - 450*x^8 + 935*x^7 - 737*x^6 - 670*x^5 + 1953*x^4 + 601*x^3 - 913*x^2 - 428*x - 43)
\( x^{15} - 3 x^{14} - 8 x^{13} + 43 x^{12} - 98 x^{11} + 206 x^{10} - 138 x^{9} - 450 x^{8} + 935 x^{7} + \cdots - 43 \)
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: | $[9, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-13498002330014577328128\)
\(\medspace = -\,2^{18}\cdot 3^{12}\cdot 7^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(29.88\) | 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: | $2^{3/2}3^{4/5}7^{11/12}\approx 40.54293644764779$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{24\!\cdots\!81}a^{14}+\frac{82\!\cdots\!94}{24\!\cdots\!81}a^{13}-\frac{11\!\cdots\!65}{24\!\cdots\!81}a^{12}-\frac{85\!\cdots\!83}{24\!\cdots\!81}a^{11}-\frac{99\!\cdots\!26}{24\!\cdots\!81}a^{10}+\frac{43\!\cdots\!85}{24\!\cdots\!81}a^{9}-\frac{11\!\cdots\!87}{24\!\cdots\!81}a^{8}-\frac{12\!\cdots\!13}{24\!\cdots\!81}a^{7}-\frac{12\!\cdots\!08}{24\!\cdots\!81}a^{6}-\frac{79\!\cdots\!01}{24\!\cdots\!81}a^{5}-\frac{49\!\cdots\!02}{24\!\cdots\!81}a^{4}+\frac{12\!\cdots\!45}{24\!\cdots\!81}a^{3}-\frac{62\!\cdots\!94}{24\!\cdots\!81}a^{2}+\frac{78\!\cdots\!93}{24\!\cdots\!81}a-\frac{99\!\cdots\!15}{56\!\cdots\!67}$
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
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: | $11$ | 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: |
$\frac{11\!\cdots\!99}{11\!\cdots\!89}a^{14}-\frac{35\!\cdots\!50}{11\!\cdots\!89}a^{13}-\frac{80\!\cdots\!10}{11\!\cdots\!89}a^{12}+\frac{49\!\cdots\!50}{11\!\cdots\!89}a^{11}-\frac{11\!\cdots\!50}{11\!\cdots\!89}a^{10}+\frac{25\!\cdots\!30}{11\!\cdots\!89}a^{9}-\frac{21\!\cdots\!37}{11\!\cdots\!89}a^{8}-\frac{43\!\cdots\!10}{11\!\cdots\!89}a^{7}+\frac{11\!\cdots\!83}{11\!\cdots\!89}a^{6}-\frac{10\!\cdots\!26}{11\!\cdots\!89}a^{5}-\frac{53\!\cdots\!47}{11\!\cdots\!89}a^{4}+\frac{22\!\cdots\!98}{11\!\cdots\!89}a^{3}+\frac{20\!\cdots\!33}{11\!\cdots\!89}a^{2}-\frac{11\!\cdots\!02}{11\!\cdots\!89}a-\frac{27\!\cdots\!79}{26\!\cdots\!23}$, $\frac{11\!\cdots\!99}{11\!\cdots\!89}a^{14}+\frac{35\!\cdots\!50}{11\!\cdots\!89}a^{13}+\frac{80\!\cdots\!10}{11\!\cdots\!89}a^{12}-\frac{49\!\cdots\!50}{11\!\cdots\!89}a^{11}+\frac{11\!\cdots\!50}{11\!\cdots\!89}a^{10}-\frac{25\!\cdots\!30}{11\!\cdots\!89}a^{9}+\frac{21\!\cdots\!37}{11\!\cdots\!89}a^{8}+\frac{43\!\cdots\!10}{11\!\cdots\!89}a^{7}-\frac{11\!\cdots\!83}{11\!\cdots\!89}a^{6}+\frac{10\!\cdots\!26}{11\!\cdots\!89}a^{5}+\frac{53\!\cdots\!47}{11\!\cdots\!89}a^{4}-\frac{22\!\cdots\!98}{11\!\cdots\!89}a^{3}-\frac{20\!\cdots\!33}{11\!\cdots\!89}a^{2}+\frac{11\!\cdots\!02}{11\!\cdots\!89}a+\frac{54\!\cdots\!02}{26\!\cdots\!23}$, $\frac{35\!\cdots\!24}{24\!\cdots\!81}a^{14}+\frac{13\!\cdots\!39}{24\!\cdots\!81}a^{13}+\frac{17\!\cdots\!18}{24\!\cdots\!81}a^{12}-\frac{16\!\cdots\!50}{24\!\cdots\!81}a^{11}+\frac{47\!\cdots\!00}{24\!\cdots\!81}a^{10}-\frac{10\!\cdots\!63}{24\!\cdots\!81}a^{9}+\frac{13\!\cdots\!89}{24\!\cdots\!81}a^{8}+\frac{55\!\cdots\!37}{24\!\cdots\!81}a^{7}-\frac{37\!\cdots\!39}{24\!\cdots\!81}a^{6}+\frac{55\!\cdots\!60}{24\!\cdots\!81}a^{5}-\frac{18\!\cdots\!86}{24\!\cdots\!81}a^{4}-\frac{56\!\cdots\!13}{24\!\cdots\!81}a^{3}+\frac{26\!\cdots\!80}{24\!\cdots\!81}a^{2}+\frac{13\!\cdots\!31}{24\!\cdots\!81}a+\frac{43\!\cdots\!37}{56\!\cdots\!67}$, $\frac{21\!\cdots\!61}{24\!\cdots\!81}a^{14}+\frac{77\!\cdots\!92}{24\!\cdots\!81}a^{13}+\frac{12\!\cdots\!70}{24\!\cdots\!81}a^{12}-\frac{99\!\cdots\!33}{24\!\cdots\!81}a^{11}+\frac{27\!\cdots\!58}{24\!\cdots\!81}a^{10}-\frac{60\!\cdots\!15}{24\!\cdots\!81}a^{9}+\frac{67\!\cdots\!03}{24\!\cdots\!81}a^{8}+\frac{53\!\cdots\!22}{24\!\cdots\!81}a^{7}-\frac{23\!\cdots\!79}{24\!\cdots\!81}a^{6}+\frac{30\!\cdots\!96}{24\!\cdots\!81}a^{5}-\frac{45\!\cdots\!94}{24\!\cdots\!81}a^{4}-\frac{38\!\cdots\!19}{24\!\cdots\!81}a^{3}+\frac{10\!\cdots\!94}{24\!\cdots\!81}a^{2}+\frac{12\!\cdots\!97}{24\!\cdots\!81}a+\frac{39\!\cdots\!30}{56\!\cdots\!67}$, $\frac{78\!\cdots\!79}{24\!\cdots\!81}a^{14}-\frac{27\!\cdots\!76}{24\!\cdots\!81}a^{13}-\frac{49\!\cdots\!52}{24\!\cdots\!81}a^{12}+\frac{36\!\cdots\!47}{24\!\cdots\!81}a^{11}-\frac{95\!\cdots\!89}{24\!\cdots\!81}a^{10}+\frac{21\!\cdots\!46}{24\!\cdots\!81}a^{9}-\frac{21\!\cdots\!02}{24\!\cdots\!81}a^{8}-\frac{24\!\cdots\!61}{24\!\cdots\!81}a^{7}+\frac{86\!\cdots\!18}{24\!\cdots\!81}a^{6}-\frac{10\!\cdots\!57}{24\!\cdots\!81}a^{5}+\frac{25\!\cdots\!71}{24\!\cdots\!81}a^{4}+\frac{14\!\cdots\!10}{24\!\cdots\!81}a^{3}-\frac{28\!\cdots\!81}{24\!\cdots\!81}a^{2}-\frac{52\!\cdots\!65}{24\!\cdots\!81}a-\frac{16\!\cdots\!45}{56\!\cdots\!67}$, $\frac{49\!\cdots\!30}{24\!\cdots\!81}a^{14}-\frac{15\!\cdots\!86}{24\!\cdots\!81}a^{13}-\frac{37\!\cdots\!58}{24\!\cdots\!81}a^{12}+\frac{22\!\cdots\!48}{24\!\cdots\!81}a^{11}-\frac{52\!\cdots\!40}{24\!\cdots\!81}a^{10}+\frac{10\!\cdots\!32}{24\!\cdots\!81}a^{9}-\frac{82\!\cdots\!34}{24\!\cdots\!81}a^{8}-\frac{21\!\cdots\!10}{24\!\cdots\!81}a^{7}+\frac{51\!\cdots\!83}{24\!\cdots\!81}a^{6}-\frac{43\!\cdots\!98}{24\!\cdots\!81}a^{5}-\frac{29\!\cdots\!22}{24\!\cdots\!81}a^{4}+\frac{10\!\cdots\!52}{24\!\cdots\!81}a^{3}+\frac{12\!\cdots\!37}{24\!\cdots\!81}a^{2}-\frac{58\!\cdots\!22}{24\!\cdots\!81}a-\frac{38\!\cdots\!44}{56\!\cdots\!67}$, $\frac{26\!\cdots\!94}{24\!\cdots\!81}a^{14}+\frac{12\!\cdots\!97}{24\!\cdots\!81}a^{13}+\frac{30\!\cdots\!59}{24\!\cdots\!81}a^{12}-\frac{13\!\cdots\!14}{24\!\cdots\!81}a^{11}+\frac{47\!\cdots\!08}{24\!\cdots\!81}a^{10}-\frac{12\!\cdots\!74}{24\!\cdots\!81}a^{9}+\frac{19\!\cdots\!88}{24\!\cdots\!81}a^{8}-\frac{10\!\cdots\!17}{24\!\cdots\!81}a^{7}-\frac{24\!\cdots\!16}{24\!\cdots\!81}a^{6}+\frac{64\!\cdots\!43}{24\!\cdots\!81}a^{5}-\frac{61\!\cdots\!85}{24\!\cdots\!81}a^{4}-\frac{40\!\cdots\!12}{24\!\cdots\!81}a^{3}+\frac{32\!\cdots\!73}{24\!\cdots\!81}a^{2}-\frac{87\!\cdots\!03}{24\!\cdots\!81}a-\frac{85\!\cdots\!73}{56\!\cdots\!67}$, $\frac{11\!\cdots\!14}{24\!\cdots\!81}a^{14}-\frac{40\!\cdots\!82}{24\!\cdots\!81}a^{13}-\frac{64\!\cdots\!51}{24\!\cdots\!81}a^{12}+\frac{52\!\cdots\!27}{24\!\cdots\!81}a^{11}-\frac{14\!\cdots\!46}{24\!\cdots\!81}a^{10}+\frac{31\!\cdots\!42}{24\!\cdots\!81}a^{9}-\frac{34\!\cdots\!05}{24\!\cdots\!81}a^{8}-\frac{29\!\cdots\!77}{24\!\cdots\!81}a^{7}+\frac{12\!\cdots\!22}{24\!\cdots\!81}a^{6}-\frac{15\!\cdots\!72}{24\!\cdots\!81}a^{5}+\frac{16\!\cdots\!22}{24\!\cdots\!81}a^{4}+\frac{20\!\cdots\!25}{24\!\cdots\!81}a^{3}-\frac{51\!\cdots\!53}{24\!\cdots\!81}a^{2}-\frac{79\!\cdots\!82}{24\!\cdots\!81}a-\frac{32\!\cdots\!78}{56\!\cdots\!67}$, $\frac{11\!\cdots\!72}{24\!\cdots\!81}a^{14}+\frac{38\!\cdots\!23}{24\!\cdots\!81}a^{13}+\frac{78\!\cdots\!31}{24\!\cdots\!81}a^{12}-\frac{51\!\cdots\!56}{24\!\cdots\!81}a^{11}+\frac{12\!\cdots\!63}{24\!\cdots\!81}a^{10}-\frac{27\!\cdots\!97}{24\!\cdots\!81}a^{9}+\frac{24\!\cdots\!97}{24\!\cdots\!81}a^{8}+\frac{43\!\cdots\!29}{24\!\cdots\!81}a^{7}-\frac{12\!\cdots\!00}{24\!\cdots\!81}a^{6}+\frac{12\!\cdots\!69}{24\!\cdots\!81}a^{5}+\frac{37\!\cdots\!41}{24\!\cdots\!81}a^{4}-\frac{23\!\cdots\!02}{24\!\cdots\!81}a^{3}+\frac{11\!\cdots\!52}{24\!\cdots\!81}a^{2}+\frac{10\!\cdots\!60}{24\!\cdots\!81}a+\frac{33\!\cdots\!48}{56\!\cdots\!67}$, $\frac{56\!\cdots\!79}{11\!\cdots\!89}a^{14}-\frac{18\!\cdots\!30}{11\!\cdots\!89}a^{13}-\frac{37\!\cdots\!59}{11\!\cdots\!89}a^{12}+\frac{25\!\cdots\!46}{11\!\cdots\!89}a^{11}-\frac{65\!\cdots\!16}{11\!\cdots\!89}a^{10}+\frac{13\!\cdots\!24}{11\!\cdots\!89}a^{9}-\frac{12\!\cdots\!25}{11\!\cdots\!89}a^{8}-\frac{21\!\cdots\!12}{11\!\cdots\!89}a^{7}+\frac{62\!\cdots\!77}{11\!\cdots\!89}a^{6}-\frac{62\!\cdots\!01}{11\!\cdots\!89}a^{5}-\frac{20\!\cdots\!46}{11\!\cdots\!89}a^{4}+\frac{12\!\cdots\!66}{11\!\cdots\!89}a^{3}-\frac{14\!\cdots\!41}{11\!\cdots\!89}a^{2}-\frac{57\!\cdots\!76}{11\!\cdots\!89}a-\frac{35\!\cdots\!32}{26\!\cdots\!23}$, $\frac{16\!\cdots\!67}{24\!\cdots\!81}a^{14}-\frac{58\!\cdots\!03}{24\!\cdots\!81}a^{13}-\frac{88\!\cdots\!31}{24\!\cdots\!81}a^{12}+\frac{73\!\cdots\!13}{24\!\cdots\!81}a^{11}-\frac{21\!\cdots\!54}{24\!\cdots\!81}a^{10}+\frac{49\!\cdots\!67}{24\!\cdots\!81}a^{9}-\frac{60\!\cdots\!93}{24\!\cdots\!81}a^{8}-\frac{20\!\cdots\!84}{24\!\cdots\!81}a^{7}+\frac{15\!\cdots\!86}{24\!\cdots\!81}a^{6}-\frac{23\!\cdots\!79}{24\!\cdots\!81}a^{5}+\frac{10\!\cdots\!03}{24\!\cdots\!81}a^{4}+\frac{19\!\cdots\!52}{24\!\cdots\!81}a^{3}-\frac{63\!\cdots\!83}{24\!\cdots\!81}a^{2}-\frac{61\!\cdots\!19}{24\!\cdots\!81}a-\frac{12\!\cdots\!37}{56\!\cdots\!67}$
| 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: | \( 654074.2800450686 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{3}\cdot 654074.2800450686 \cdot 1}{2\cdot\sqrt{13498002330014577328128}}\cr\approx \mathstrut & 0.357496581132441 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 8*x^13 + 43*x^12 - 98*x^11 + 206*x^10 - 138*x^9 - 450*x^8 + 935*x^7 - 737*x^6 - 670*x^5 + 1953*x^4 + 601*x^3 - 913*x^2 - 428*x - 43)
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 - 3*x^14 - 8*x^13 + 43*x^12 - 98*x^11 + 206*x^10 - 138*x^9 - 450*x^8 + 935*x^7 - 737*x^6 - 670*x^5 + 1953*x^4 + 601*x^3 - 913*x^2 - 428*x - 43, 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 - 3*x^14 - 8*x^13 + 43*x^12 - 98*x^11 + 206*x^10 - 138*x^9 - 450*x^8 + 935*x^7 - 737*x^6 - 670*x^5 + 1953*x^4 + 601*x^3 - 913*x^2 - 428*x - 43);
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 - 3*x^14 - 8*x^13 + 43*x^12 - 98*x^11 + 206*x^10 - 138*x^9 - 450*x^8 + 935*x^7 - 737*x^6 - 670*x^5 + 1953*x^4 + 601*x^3 - 913*x^2 - 428*x - 43);
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
$C_3\times S_5$ (as 15T24):
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 360 |
| The 21 conjugacy class representatives for $S_5 \times C_3$ |
| Character table for $S_5 \times C_3$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 5.3.1778112.1 |
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 sibling: | 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 | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{3}$ | R | $15$ | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.5.0.1}{5} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{3}$ | $15$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.3.0.1}{3} }^{5}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.12.18.59 | $x^{12} + 6 x^{11} + 22 x^{10} + 56 x^{9} + 126 x^{8} + 240 x^{7} + 332 x^{6} - 18 x^{5} - 459 x^{4} - 394 x^{3} - 344 x^{2} + 138 x + 423$ | $4$ | $3$ | $18$ | $A_4$ | $[2, 2]^{3}$ | |
|
\(3\)
| 3.15.12.1 | $x^{15} + 10 x^{13} + 5 x^{12} + 40 x^{11} + 49 x^{10} + 90 x^{9} + 30 x^{8} - 130 x^{7} + 410 x^{6} + 269 x^{5} + 765 x^{4} - 515 x^{3} + 730 x^{2} + 205 x + 94$ | $5$ | $3$ | $12$ | $F_5\times C_3$ | $[\ ]_{5}^{12}$ |
|
\(7\)
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.12.11.4 | $x^{12} + 42$ | $12$ | $1$ | $11$ | $D_4 \times C_3$ | $[\ ]_{12}^{2}$ |