Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 11*x^13 + 11*x^12 - 147*x^11 + 205*x^10 - 188*x^8 - 988*x^7 + 729*x^6 + 1598*x^5 - 1876*x^4 + 302*x^3 + 325*x^2 - 126*x + 7)
gp: K = bnfinit(y^16 - 4*y^15 + 10*y^14 - 11*y^13 + 11*y^12 - 147*y^11 + 205*y^10 - 188*y^8 - 988*y^7 + 729*y^6 + 1598*y^5 - 1876*y^4 + 302*y^3 + 325*y^2 - 126*y + 7, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 10*x^14 - 11*x^13 + 11*x^12 - 147*x^11 + 205*x^10 - 188*x^8 - 988*x^7 + 729*x^6 + 1598*x^5 - 1876*x^4 + 302*x^3 + 325*x^2 - 126*x + 7);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 10*x^14 - 11*x^13 + 11*x^12 - 147*x^11 + 205*x^10 - 188*x^8 - 988*x^7 + 729*x^6 + 1598*x^5 - 1876*x^4 + 302*x^3 + 325*x^2 - 126*x + 7)
\( x^{16} - 4 x^{15} + 10 x^{14} - 11 x^{13} + 11 x^{12} - 147 x^{11} + 205 x^{10} - 188 x^{8} - 988 x^{7} + \cdots + 7 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(4328092617651142144697025\)
\(\medspace = 3^{8}\cdot 5^{2}\cdot 7^{2}\cdot 53^{2}\cdot 61^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(34.66\) | 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: | $3^{1/2}5^{1/2}7^{1/2}53^{1/2}61^{2/3}\approx 1155.985733589208$ | ||
| Ramified primes: |
\(3\), \(5\), \(7\), \(53\), \(61\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{10}a^{14}-\frac{1}{5}a^{13}-\frac{1}{10}a^{12}+\frac{1}{10}a^{11}-\frac{2}{5}a^{9}-\frac{3}{10}a^{8}+\frac{1}{5}a^{7}-\frac{3}{10}a^{6}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a+\frac{1}{10}$, $\frac{1}{11\!\cdots\!80}a^{15}-\frac{28\!\cdots\!41}{11\!\cdots\!80}a^{14}+\frac{16\!\cdots\!67}{11\!\cdots\!80}a^{13}+\frac{29\!\cdots\!19}{11\!\cdots\!98}a^{12}+\frac{18\!\cdots\!81}{11\!\cdots\!80}a^{11}-\frac{10\!\cdots\!41}{28\!\cdots\!45}a^{10}+\frac{21\!\cdots\!53}{11\!\cdots\!80}a^{9}-\frac{47\!\cdots\!61}{11\!\cdots\!80}a^{8}+\frac{42\!\cdots\!89}{11\!\cdots\!80}a^{7}+\frac{12\!\cdots\!59}{11\!\cdots\!80}a^{6}+\frac{24\!\cdots\!53}{56\!\cdots\!90}a^{5}-\frac{73\!\cdots\!61}{28\!\cdots\!45}a^{4}+\frac{20\!\cdots\!08}{28\!\cdots\!45}a^{3}+\frac{87\!\cdots\!09}{56\!\cdots\!90}a^{2}+\frac{69\!\cdots\!23}{15\!\cdots\!60}a+\frac{19\!\cdots\!51}{11\!\cdots\!80}$
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: | $9$ | 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: |
$\frac{43\!\cdots\!77}{11\!\cdots\!80}a^{15}-\frac{52\!\cdots\!49}{11\!\cdots\!80}a^{14}+\frac{98\!\cdots\!23}{11\!\cdots\!80}a^{13}+\frac{12\!\cdots\!81}{56\!\cdots\!90}a^{12}+\frac{55\!\cdots\!17}{22\!\cdots\!96}a^{11}-\frac{14\!\cdots\!72}{28\!\cdots\!45}a^{10}-\frac{77\!\cdots\!11}{11\!\cdots\!80}a^{9}+\frac{35\!\cdots\!79}{11\!\cdots\!80}a^{8}+\frac{77\!\cdots\!09}{11\!\cdots\!80}a^{7}-\frac{54\!\cdots\!41}{11\!\cdots\!80}a^{6}-\frac{53\!\cdots\!91}{56\!\cdots\!90}a^{5}-\frac{13\!\cdots\!74}{28\!\cdots\!45}a^{4}+\frac{55\!\cdots\!11}{56\!\cdots\!49}a^{3}+\frac{10\!\cdots\!43}{56\!\cdots\!90}a^{2}-\frac{85\!\cdots\!81}{15\!\cdots\!60}a+\frac{16\!\cdots\!91}{22\!\cdots\!96}$, $\frac{50\!\cdots\!19}{11\!\cdots\!80}a^{15}-\frac{14\!\cdots\!13}{11\!\cdots\!80}a^{14}+\frac{33\!\cdots\!41}{11\!\cdots\!80}a^{13}-\frac{41\!\cdots\!14}{28\!\cdots\!45}a^{12}+\frac{62\!\cdots\!61}{22\!\cdots\!96}a^{11}-\frac{17\!\cdots\!84}{28\!\cdots\!45}a^{10}+\frac{26\!\cdots\!83}{11\!\cdots\!80}a^{9}+\frac{44\!\cdots\!23}{11\!\cdots\!80}a^{8}-\frac{48\!\cdots\!97}{11\!\cdots\!80}a^{7}-\frac{56\!\cdots\!97}{11\!\cdots\!80}a^{6}-\frac{11\!\cdots\!47}{56\!\cdots\!90}a^{5}+\frac{17\!\cdots\!72}{28\!\cdots\!45}a^{4}-\frac{61\!\cdots\!89}{56\!\cdots\!49}a^{3}-\frac{52\!\cdots\!89}{56\!\cdots\!90}a^{2}+\frac{45\!\cdots\!73}{15\!\cdots\!60}a-\frac{20\!\cdots\!65}{22\!\cdots\!96}$, $\frac{86\!\cdots\!09}{11\!\cdots\!80}a^{15}-\frac{16\!\cdots\!13}{11\!\cdots\!80}a^{14}+\frac{30\!\cdots\!71}{11\!\cdots\!80}a^{13}+\frac{16\!\cdots\!17}{56\!\cdots\!90}a^{12}+\frac{45\!\cdots\!41}{22\!\cdots\!96}a^{11}-\frac{28\!\cdots\!29}{28\!\cdots\!45}a^{10}-\frac{80\!\cdots\!27}{11\!\cdots\!80}a^{9}+\frac{12\!\cdots\!83}{11\!\cdots\!80}a^{8}+\frac{66\!\cdots\!73}{11\!\cdots\!80}a^{7}-\frac{10\!\cdots\!57}{11\!\cdots\!80}a^{6}-\frac{70\!\cdots\!27}{56\!\cdots\!90}a^{5}+\frac{20\!\cdots\!17}{28\!\cdots\!45}a^{4}+\frac{57\!\cdots\!53}{56\!\cdots\!49}a^{3}-\frac{20\!\cdots\!69}{56\!\cdots\!90}a^{2}-\frac{37\!\cdots\!97}{15\!\cdots\!60}a+\frac{52\!\cdots\!95}{22\!\cdots\!96}$, $\frac{66\!\cdots\!81}{59\!\cdots\!20}a^{15}-\frac{23\!\cdots\!83}{59\!\cdots\!20}a^{14}+\frac{55\!\cdots\!91}{59\!\cdots\!20}a^{13}-\frac{11\!\cdots\!37}{14\!\cdots\!55}a^{12}+\frac{55\!\cdots\!99}{59\!\cdots\!20}a^{11}-\frac{23\!\cdots\!96}{14\!\cdots\!55}a^{10}+\frac{88\!\cdots\!21}{59\!\cdots\!20}a^{9}+\frac{67\!\cdots\!69}{11\!\cdots\!84}a^{8}-\frac{19\!\cdots\!79}{11\!\cdots\!84}a^{7}-\frac{13\!\cdots\!91}{11\!\cdots\!84}a^{6}+\frac{59\!\cdots\!31}{29\!\cdots\!10}a^{5}+\frac{25\!\cdots\!07}{14\!\cdots\!55}a^{4}-\frac{17\!\cdots\!23}{14\!\cdots\!55}a^{3}-\frac{88\!\cdots\!11}{29\!\cdots\!10}a^{2}+\frac{16\!\cdots\!31}{82\!\cdots\!40}a-\frac{30\!\cdots\!91}{59\!\cdots\!20}$, $\frac{15\!\cdots\!33}{16\!\cdots\!08}a^{15}-\frac{40\!\cdots\!89}{16\!\cdots\!08}a^{14}+\frac{87\!\cdots\!51}{16\!\cdots\!08}a^{13}-\frac{59\!\cdots\!53}{82\!\cdots\!54}a^{12}+\frac{77\!\cdots\!25}{16\!\cdots\!08}a^{11}-\frac{54\!\cdots\!88}{41\!\cdots\!27}a^{10}+\frac{866675753219625}{16\!\cdots\!08}a^{9}+\frac{17\!\cdots\!95}{16\!\cdots\!08}a^{8}-\frac{94\!\cdots\!03}{16\!\cdots\!08}a^{7}-\frac{18\!\cdots\!13}{16\!\cdots\!08}a^{6}-\frac{69\!\cdots\!33}{82\!\cdots\!54}a^{5}+\frac{47\!\cdots\!23}{41\!\cdots\!27}a^{4}+\frac{12\!\cdots\!75}{41\!\cdots\!27}a^{3}-\frac{24\!\cdots\!29}{82\!\cdots\!54}a^{2}-\frac{10\!\cdots\!41}{16\!\cdots\!08}a+\frac{54\!\cdots\!91}{16\!\cdots\!08}$, $\frac{18\!\cdots\!95}{11\!\cdots\!98}a^{15}-\frac{60\!\cdots\!55}{11\!\cdots\!98}a^{14}+\frac{14\!\cdots\!39}{11\!\cdots\!98}a^{13}-\frac{50\!\cdots\!01}{56\!\cdots\!49}a^{12}+\frac{13\!\cdots\!31}{11\!\cdots\!98}a^{11}-\frac{13\!\cdots\!88}{56\!\cdots\!49}a^{10}+\frac{18\!\cdots\!71}{11\!\cdots\!98}a^{9}+\frac{11\!\cdots\!17}{11\!\cdots\!98}a^{8}-\frac{23\!\cdots\!67}{11\!\cdots\!98}a^{7}-\frac{20\!\cdots\!27}{11\!\cdots\!98}a^{6}-\frac{75\!\cdots\!70}{56\!\cdots\!49}a^{5}+\frac{13\!\cdots\!01}{56\!\cdots\!49}a^{4}-\frac{69\!\cdots\!20}{56\!\cdots\!49}a^{3}-\frac{57\!\cdots\!43}{56\!\cdots\!49}a^{2}+\frac{36\!\cdots\!35}{15\!\cdots\!26}a-\frac{61\!\cdots\!83}{11\!\cdots\!98}$, $\frac{39\!\cdots\!25}{22\!\cdots\!96}a^{15}-\frac{11\!\cdots\!69}{22\!\cdots\!96}a^{14}+\frac{24\!\cdots\!79}{22\!\cdots\!96}a^{13}-\frac{47\!\cdots\!73}{11\!\cdots\!98}a^{12}+\frac{22\!\cdots\!53}{22\!\cdots\!96}a^{11}-\frac{13\!\cdots\!99}{56\!\cdots\!49}a^{10}+\frac{13\!\cdots\!77}{22\!\cdots\!96}a^{9}+\frac{37\!\cdots\!71}{22\!\cdots\!96}a^{8}-\frac{32\!\cdots\!71}{22\!\cdots\!96}a^{7}-\frac{44\!\cdots\!81}{22\!\cdots\!96}a^{6}-\frac{11\!\cdots\!77}{11\!\cdots\!98}a^{5}+\frac{12\!\cdots\!34}{56\!\cdots\!49}a^{4}-\frac{50\!\cdots\!15}{56\!\cdots\!49}a^{3}-\frac{47\!\cdots\!85}{11\!\cdots\!98}a^{2}+\frac{11\!\cdots\!07}{31\!\cdots\!52}a-\frac{67\!\cdots\!57}{22\!\cdots\!96}$, $\frac{21\!\cdots\!69}{22\!\cdots\!96}a^{15}-\frac{35\!\cdots\!81}{11\!\cdots\!80}a^{14}+\frac{93\!\cdots\!27}{11\!\cdots\!80}a^{13}-\frac{49\!\cdots\!67}{56\!\cdots\!90}a^{12}+\frac{16\!\cdots\!29}{11\!\cdots\!80}a^{11}-\frac{78\!\cdots\!90}{56\!\cdots\!49}a^{10}+\frac{13\!\cdots\!69}{11\!\cdots\!80}a^{9}-\frac{65\!\cdots\!77}{11\!\cdots\!80}a^{8}-\frac{11\!\cdots\!07}{11\!\cdots\!80}a^{7}-\frac{11\!\cdots\!17}{11\!\cdots\!80}a^{6}-\frac{12\!\cdots\!91}{56\!\cdots\!90}a^{5}+\frac{17\!\cdots\!04}{28\!\cdots\!45}a^{4}-\frac{17\!\cdots\!33}{28\!\cdots\!45}a^{3}+\frac{12\!\cdots\!01}{11\!\cdots\!98}a^{2}-\frac{10\!\cdots\!81}{15\!\cdots\!60}a+\frac{82\!\cdots\!19}{11\!\cdots\!80}$, $\frac{26\!\cdots\!01}{28\!\cdots\!45}a^{15}-\frac{16\!\cdots\!11}{56\!\cdots\!90}a^{14}+\frac{18\!\cdots\!81}{28\!\cdots\!45}a^{13}-\frac{18\!\cdots\!61}{56\!\cdots\!90}a^{12}+\frac{22\!\cdots\!73}{56\!\cdots\!90}a^{11}-\frac{36\!\cdots\!19}{28\!\cdots\!45}a^{10}+\frac{24\!\cdots\!11}{28\!\cdots\!45}a^{9}+\frac{10\!\cdots\!33}{11\!\cdots\!98}a^{8}-\frac{77\!\cdots\!41}{56\!\cdots\!49}a^{7}-\frac{11\!\cdots\!57}{11\!\cdots\!98}a^{6}+\frac{17\!\cdots\!87}{28\!\cdots\!45}a^{5}+\frac{41\!\cdots\!58}{28\!\cdots\!45}a^{4}-\frac{17\!\cdots\!27}{28\!\cdots\!45}a^{3}-\frac{66\!\cdots\!87}{28\!\cdots\!45}a^{2}+\frac{48\!\cdots\!31}{39\!\cdots\!65}a-\frac{38\!\cdots\!87}{56\!\cdots\!90}$
| 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: | \( 2759124.66733 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 2759124.66733 \cdot 1}{2\cdot\sqrt{4328092617651142144697025}}\cr\approx \mathstrut & 0.652818191788 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 11*x^13 + 11*x^12 - 147*x^11 + 205*x^10 - 188*x^8 - 988*x^7 + 729*x^6 + 1598*x^5 - 1876*x^4 + 302*x^3 + 325*x^2 - 126*x + 7)
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^16 - 4*x^15 + 10*x^14 - 11*x^13 + 11*x^12 - 147*x^11 + 205*x^10 - 188*x^8 - 988*x^7 + 729*x^6 + 1598*x^5 - 1876*x^4 + 302*x^3 + 325*x^2 - 126*x + 7, 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^16 - 4*x^15 + 10*x^14 - 11*x^13 + 11*x^12 - 147*x^11 + 205*x^10 - 188*x^8 - 988*x^7 + 729*x^6 + 1598*x^5 - 1876*x^4 + 302*x^3 + 325*x^2 - 126*x + 7);
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^16 - 4*x^15 + 10*x^14 - 11*x^13 + 11*x^12 - 147*x^11 + 205*x^10 - 188*x^8 - 988*x^7 + 729*x^6 + 1598*x^5 - 1876*x^4 + 302*x^3 + 325*x^2 - 126*x + 7);
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
$C_2\wr Q_8.A_4$ (as 16T1811):
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 24576 |
| The 88 conjugacy class representatives for $C_2\wr Q_8.A_4$ |
| Character table for $C_2\wr Q_8.A_4$ |
Intermediate fields
| 4.4.33489.1, 8.6.39252959235.1 |
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 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 16.6.347360347513773093841541235.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.3.0.1}{3} }^{2}$ | R | R | R | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | R | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.16.8.1 | $x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(7\)
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
|
\(53\)
| 53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 53.4.2.1 | $x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 53.4.0.1 | $x^{4} + 9 x^{2} + 38 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 53.4.0.1 | $x^{4} + 9 x^{2} + 38 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(61\)
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.12.8.1 | $x^{12} + 9 x^{10} + 364 x^{9} + 33 x^{8} + 720 x^{7} - 16731 x^{6} - 1002 x^{5} - 64041 x^{4} - 1125646 x^{3} + 428523 x^{2} - 4549998 x + 62837938$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |