Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 36*x^16 - 124*x^15 + 667*x^14 - 2334*x^13 + 8164*x^12 - 20338*x^11 + 54321*x^10 - 96972*x^9 + 150518*x^8 - 244982*x^7 + 834376*x^6 - 2158328*x^5 + 4885022*x^4 - 8338866*x^3 + 13279313*x^2 - 14690014*x + 10525279)
gp: K = bnfinit(y^18 - 2*y^17 + 36*y^16 - 124*y^15 + 667*y^14 - 2334*y^13 + 8164*y^12 - 20338*y^11 + 54321*y^10 - 96972*y^9 + 150518*y^8 - 244982*y^7 + 834376*y^6 - 2158328*y^5 + 4885022*y^4 - 8338866*y^3 + 13279313*y^2 - 14690014*y + 10525279, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 + 36*x^16 - 124*x^15 + 667*x^14 - 2334*x^13 + 8164*x^12 - 20338*x^11 + 54321*x^10 - 96972*x^9 + 150518*x^8 - 244982*x^7 + 834376*x^6 - 2158328*x^5 + 4885022*x^4 - 8338866*x^3 + 13279313*x^2 - 14690014*x + 10525279);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 2*x^17 + 36*x^16 - 124*x^15 + 667*x^14 - 2334*x^13 + 8164*x^12 - 20338*x^11 + 54321*x^10 - 96972*x^9 + 150518*x^8 - 244982*x^7 + 834376*x^6 - 2158328*x^5 + 4885022*x^4 - 8338866*x^3 + 13279313*x^2 - 14690014*x + 10525279)
\( x^{18} - 2 x^{17} + 36 x^{16} - 124 x^{15} + 667 x^{14} - 2334 x^{13} + 8164 x^{12} - 20338 x^{11} + \cdots + 10525279 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-3830034706318154794675914145792\)
\(\medspace = -\,2^{18}\cdot 19^{16}\cdot 37^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(50.01\) | 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: | $2^{7/4}19^{8/9}37^{1/2}\approx 280.26781127800797$ | ||
| Ramified primes: |
\(2\), \(19\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-37}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{256}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{42\!\cdots\!49}a^{17}+\frac{23\!\cdots\!35}{42\!\cdots\!49}a^{16}+\frac{12\!\cdots\!09}{42\!\cdots\!49}a^{15}+\frac{13\!\cdots\!74}{42\!\cdots\!49}a^{14}+\frac{18\!\cdots\!76}{42\!\cdots\!49}a^{13}+\frac{25\!\cdots\!92}{42\!\cdots\!49}a^{12}-\frac{17\!\cdots\!07}{42\!\cdots\!49}a^{11}+\frac{15\!\cdots\!44}{28\!\cdots\!99}a^{10}-\frac{11\!\cdots\!01}{42\!\cdots\!49}a^{9}+\frac{75\!\cdots\!68}{42\!\cdots\!49}a^{8}+\frac{94\!\cdots\!93}{42\!\cdots\!49}a^{7}-\frac{15\!\cdots\!05}{42\!\cdots\!49}a^{6}+\frac{23\!\cdots\!91}{42\!\cdots\!49}a^{5}+\frac{18\!\cdots\!76}{42\!\cdots\!49}a^{4}-\frac{20\!\cdots\!27}{42\!\cdots\!49}a^{3}+\frac{83\!\cdots\!83}{42\!\cdots\!49}a^{2}-\frac{20\!\cdots\!32}{42\!\cdots\!49}a-\frac{28\!\cdots\!50}{11\!\cdots\!77}$
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
$C_{2}\times C_{2}\times C_{570}$, which has order $2280$ (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: |
$\frac{72\!\cdots\!38}{42\!\cdots\!49}a^{17}-\frac{29\!\cdots\!69}{42\!\cdots\!49}a^{16}+\frac{24\!\cdots\!14}{42\!\cdots\!49}a^{15}-\frac{14\!\cdots\!52}{42\!\cdots\!49}a^{14}+\frac{49\!\cdots\!00}{42\!\cdots\!49}a^{13}-\frac{25\!\cdots\!26}{42\!\cdots\!49}a^{12}+\frac{68\!\cdots\!94}{42\!\cdots\!49}a^{11}-\frac{14\!\cdots\!90}{28\!\cdots\!99}a^{10}+\frac{45\!\cdots\!52}{42\!\cdots\!49}a^{9}-\frac{10\!\cdots\!16}{42\!\cdots\!49}a^{8}+\frac{96\!\cdots\!08}{42\!\cdots\!49}a^{7}-\frac{27\!\cdots\!43}{42\!\cdots\!49}a^{6}+\frac{50\!\cdots\!60}{42\!\cdots\!49}a^{5}-\frac{22\!\cdots\!72}{42\!\cdots\!49}a^{4}+\frac{46\!\cdots\!96}{42\!\cdots\!49}a^{3}-\frac{89\!\cdots\!83}{42\!\cdots\!49}a^{2}+\frac{95\!\cdots\!18}{42\!\cdots\!49}a-\frac{43\!\cdots\!53}{11\!\cdots\!77}$, $\frac{93\!\cdots\!44}{42\!\cdots\!49}a^{17}-\frac{23\!\cdots\!81}{42\!\cdots\!49}a^{16}+\frac{30\!\cdots\!86}{42\!\cdots\!49}a^{15}-\frac{13\!\cdots\!56}{42\!\cdots\!49}a^{14}+\frac{55\!\cdots\!94}{42\!\cdots\!49}a^{13}-\frac{22\!\cdots\!98}{42\!\cdots\!49}a^{12}+\frac{70\!\cdots\!00}{42\!\cdots\!49}a^{11}-\frac{12\!\cdots\!85}{28\!\cdots\!99}a^{10}+\frac{45\!\cdots\!16}{42\!\cdots\!49}a^{9}-\frac{91\!\cdots\!95}{42\!\cdots\!49}a^{8}+\frac{11\!\cdots\!24}{42\!\cdots\!49}a^{7}-\frac{27\!\cdots\!74}{42\!\cdots\!49}a^{6}+\frac{91\!\cdots\!76}{42\!\cdots\!49}a^{5}-\frac{21\!\cdots\!01}{42\!\cdots\!49}a^{4}+\frac{36\!\cdots\!88}{42\!\cdots\!49}a^{3}-\frac{64\!\cdots\!57}{42\!\cdots\!49}a^{2}+\frac{83\!\cdots\!36}{42\!\cdots\!49}a-\frac{26\!\cdots\!07}{11\!\cdots\!77}$, $\frac{93\!\cdots\!38}{42\!\cdots\!49}a^{17}-\frac{17\!\cdots\!86}{42\!\cdots\!49}a^{16}+\frac{29\!\cdots\!14}{42\!\cdots\!49}a^{15}-\frac{11\!\cdots\!13}{42\!\cdots\!49}a^{14}+\frac{45\!\cdots\!56}{42\!\cdots\!49}a^{13}-\frac{20\!\cdots\!95}{42\!\cdots\!49}a^{12}+\frac{51\!\cdots\!90}{42\!\cdots\!49}a^{11}-\frac{96\!\cdots\!64}{28\!\cdots\!99}a^{10}+\frac{28\!\cdots\!72}{42\!\cdots\!49}a^{9}-\frac{58\!\cdots\!75}{42\!\cdots\!49}a^{8}+\frac{77\!\cdots\!00}{42\!\cdots\!49}a^{7}-\frac{16\!\cdots\!81}{42\!\cdots\!49}a^{6}+\frac{42\!\cdots\!20}{42\!\cdots\!49}a^{5}-\frac{15\!\cdots\!35}{42\!\cdots\!49}a^{4}+\frac{22\!\cdots\!86}{42\!\cdots\!49}a^{3}-\frac{49\!\cdots\!30}{42\!\cdots\!49}a^{2}+\frac{61\!\cdots\!38}{42\!\cdots\!49}a-\frac{28\!\cdots\!71}{11\!\cdots\!77}$, $\frac{20\!\cdots\!36}{42\!\cdots\!49}a^{17}-\frac{37\!\cdots\!86}{42\!\cdots\!49}a^{16}+\frac{65\!\cdots\!92}{42\!\cdots\!49}a^{15}-\frac{24\!\cdots\!01}{42\!\cdots\!49}a^{14}+\frac{10\!\cdots\!46}{42\!\cdots\!49}a^{13}-\frac{42\!\cdots\!09}{42\!\cdots\!49}a^{12}+\frac{12\!\cdots\!78}{42\!\cdots\!49}a^{11}-\frac{21\!\cdots\!50}{28\!\cdots\!99}a^{10}+\frac{76\!\cdots\!10}{42\!\cdots\!49}a^{9}-\frac{14\!\cdots\!68}{42\!\cdots\!49}a^{8}+\frac{11\!\cdots\!68}{42\!\cdots\!49}a^{7}-\frac{36\!\cdots\!60}{42\!\cdots\!49}a^{6}+\frac{14\!\cdots\!82}{42\!\cdots\!49}a^{5}-\frac{35\!\cdots\!27}{42\!\cdots\!49}a^{4}+\frac{59\!\cdots\!94}{42\!\cdots\!49}a^{3}-\frac{11\!\cdots\!12}{42\!\cdots\!49}a^{2}+\frac{15\!\cdots\!50}{42\!\cdots\!49}a-\frac{38\!\cdots\!39}{11\!\cdots\!77}$, $\frac{18\!\cdots\!76}{42\!\cdots\!49}a^{17}-\frac{42\!\cdots\!00}{42\!\cdots\!49}a^{16}+\frac{60\!\cdots\!38}{42\!\cdots\!49}a^{15}-\frac{24\!\cdots\!33}{42\!\cdots\!49}a^{14}+\frac{10\!\cdots\!68}{42\!\cdots\!49}a^{13}-\frac{43\!\cdots\!51}{42\!\cdots\!49}a^{12}+\frac{13\!\cdots\!70}{42\!\cdots\!49}a^{11}-\frac{23\!\cdots\!95}{28\!\cdots\!99}a^{10}+\frac{81\!\cdots\!42}{42\!\cdots\!49}a^{9}-\frac{15\!\cdots\!48}{42\!\cdots\!49}a^{8}+\frac{14\!\cdots\!76}{42\!\cdots\!49}a^{7}-\frac{36\!\cdots\!20}{42\!\cdots\!49}a^{6}+\frac{12\!\cdots\!74}{42\!\cdots\!49}a^{5}-\frac{37\!\cdots\!43}{42\!\cdots\!49}a^{4}+\frac{70\!\cdots\!30}{42\!\cdots\!49}a^{3}-\frac{13\!\cdots\!53}{42\!\cdots\!49}a^{2}+\frac{16\!\cdots\!58}{42\!\cdots\!49}a-\frac{43\!\cdots\!37}{11\!\cdots\!77}$, $\frac{16\!\cdots\!24}{42\!\cdots\!49}a^{17}-\frac{48\!\cdots\!12}{42\!\cdots\!49}a^{16}+\frac{54\!\cdots\!32}{42\!\cdots\!49}a^{15}-\frac{25\!\cdots\!68}{42\!\cdots\!49}a^{14}+\frac{10\!\cdots\!88}{42\!\cdots\!49}a^{13}-\frac{44\!\cdots\!03}{42\!\cdots\!49}a^{12}+\frac{13\!\cdots\!16}{42\!\cdots\!49}a^{11}-\frac{23\!\cdots\!24}{28\!\cdots\!99}a^{10}+\frac{84\!\cdots\!12}{42\!\cdots\!49}a^{9}-\frac{17\!\cdots\!48}{42\!\cdots\!49}a^{8}+\frac{17\!\cdots\!20}{42\!\cdots\!49}a^{7}-\frac{41\!\cdots\!06}{42\!\cdots\!49}a^{6}+\frac{13\!\cdots\!48}{42\!\cdots\!49}a^{5}-\frac{38\!\cdots\!88}{42\!\cdots\!49}a^{4}+\frac{75\!\cdots\!80}{42\!\cdots\!49}a^{3}-\frac{14\!\cdots\!20}{42\!\cdots\!49}a^{2}+\frac{16\!\cdots\!76}{42\!\cdots\!49}a-\frac{45\!\cdots\!00}{11\!\cdots\!77}$, $\frac{18\!\cdots\!76}{42\!\cdots\!49}a^{17}-\frac{42\!\cdots\!00}{42\!\cdots\!49}a^{16}+\frac{60\!\cdots\!38}{42\!\cdots\!49}a^{15}-\frac{24\!\cdots\!33}{42\!\cdots\!49}a^{14}+\frac{10\!\cdots\!68}{42\!\cdots\!49}a^{13}-\frac{43\!\cdots\!51}{42\!\cdots\!49}a^{12}+\frac{13\!\cdots\!70}{42\!\cdots\!49}a^{11}-\frac{23\!\cdots\!95}{28\!\cdots\!99}a^{10}+\frac{81\!\cdots\!42}{42\!\cdots\!49}a^{9}-\frac{15\!\cdots\!48}{42\!\cdots\!49}a^{8}+\frac{14\!\cdots\!76}{42\!\cdots\!49}a^{7}-\frac{36\!\cdots\!20}{42\!\cdots\!49}a^{6}+\frac{12\!\cdots\!74}{42\!\cdots\!49}a^{5}-\frac{37\!\cdots\!43}{42\!\cdots\!49}a^{4}+\frac{70\!\cdots\!30}{42\!\cdots\!49}a^{3}-\frac{13\!\cdots\!53}{42\!\cdots\!49}a^{2}+\frac{16\!\cdots\!58}{42\!\cdots\!49}a-\frac{31\!\cdots\!60}{11\!\cdots\!77}$, $\frac{13\!\cdots\!90}{42\!\cdots\!49}a^{17}-\frac{27\!\cdots\!15}{42\!\cdots\!49}a^{16}+\frac{45\!\cdots\!32}{42\!\cdots\!49}a^{15}-\frac{17\!\cdots\!07}{42\!\cdots\!49}a^{14}+\frac{79\!\cdots\!38}{42\!\cdots\!49}a^{13}-\frac{31\!\cdots\!37}{42\!\cdots\!49}a^{12}+\frac{94\!\cdots\!46}{42\!\cdots\!49}a^{11}-\frac{16\!\cdots\!63}{28\!\cdots\!99}a^{10}+\frac{59\!\cdots\!20}{42\!\cdots\!49}a^{9}-\frac{11\!\cdots\!65}{42\!\cdots\!49}a^{8}+\frac{11\!\cdots\!52}{42\!\cdots\!49}a^{7}-\frac{30\!\cdots\!43}{42\!\cdots\!49}a^{6}+\frac{92\!\cdots\!52}{42\!\cdots\!49}a^{5}-\frac{27\!\cdots\!40}{42\!\cdots\!49}a^{4}+\frac{51\!\cdots\!78}{42\!\cdots\!49}a^{3}-\frac{96\!\cdots\!11}{42\!\cdots\!49}a^{2}+\frac{11\!\cdots\!10}{42\!\cdots\!49}a-\frac{31\!\cdots\!29}{11\!\cdots\!77}$
| 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: | \( 22305.8950792 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 22305.8950792 \cdot 2280}{2\cdot\sqrt{3830034706318154794675914145792}}\cr\approx \mathstrut & 0.198308791039 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 36*x^16 - 124*x^15 + 667*x^14 - 2334*x^13 + 8164*x^12 - 20338*x^11 + 54321*x^10 - 96972*x^9 + 150518*x^8 - 244982*x^7 + 834376*x^6 - 2158328*x^5 + 4885022*x^4 - 8338866*x^3 + 13279313*x^2 - 14690014*x + 10525279)
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^18 - 2*x^17 + 36*x^16 - 124*x^15 + 667*x^14 - 2334*x^13 + 8164*x^12 - 20338*x^11 + 54321*x^10 - 96972*x^9 + 150518*x^8 - 244982*x^7 + 834376*x^6 - 2158328*x^5 + 4885022*x^4 - 8338866*x^3 + 13279313*x^2 - 14690014*x + 10525279, 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^18 - 2*x^17 + 36*x^16 - 124*x^15 + 667*x^14 - 2334*x^13 + 8164*x^12 - 20338*x^11 + 54321*x^10 - 96972*x^9 + 150518*x^8 - 244982*x^7 + 834376*x^6 - 2158328*x^5 + 4885022*x^4 - 8338866*x^3 + 13279313*x^2 - 14690014*x + 10525279);
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^18 - 2*x^17 + 36*x^16 - 124*x^15 + 667*x^14 - 2334*x^13 + 8164*x^12 - 20338*x^11 + 54321*x^10 - 96972*x^9 + 150518*x^8 - 244982*x^7 + 834376*x^6 - 2158328*x^5 + 4885022*x^4 - 8338866*x^3 + 13279313*x^2 - 14690014*x + 10525279);
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
$C_2^2:C_{18}$ (as 18T26):
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 72 |
| The 24 conjugacy class representatives for $C_2^2:C_{18}$ |
| Character table for $C_2^2:C_{18}$ is not computed |
Intermediate fields
| 3.3.361.1, 6.0.308600128.1, \(\Q(\zeta_{19})^+\) |
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 36 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 | $18$ | $18$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{4}$ | $18$ | $18$ | R | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/53.9.0.1}{9} }^{2}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.18.117 | $x^{18} + 54 x^{16} + 96 x^{15} + 536 x^{14} + 4112 x^{13} - 240 x^{12} + 71360 x^{11} + 20672 x^{10} + 687040 x^{9} + 714112 x^{8} + 2937856 x^{7} + 2771072 x^{6} - 4750592 x^{5} - 12442880 x^{4} - 57267200 x^{3} - 57246976 x^{2} - 28290048 x - 139066880$ | $2$ | $9$ | $18$ | 18T26 | $[2, 2, 2]^{9}$ |
|
\(19\)
| 19.9.8.8 | $x^{9} + 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 19.9.8.8 | $x^{9} + 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |
|
\(37\)
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |