Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*x - 1)
gp: K = bnfinit(y^18 - 6*y^17 + 15*y^16 - 12*y^15 - 39*y^14 + 153*y^13 - 265*y^12 + 243*y^11 + 9*y^10 - 417*y^9 + 768*y^8 - 906*y^7 + 751*y^6 - 372*y^5 + 66*y^4 + 19*y^3 - 9*y^2 + 3*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*x - 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*x - 1)
\( x^{18} - 6 x^{17} + 15 x^{16} - 12 x^{15} - 39 x^{14} + 153 x^{13} - 265 x^{12} + 243 x^{11} + 9 x^{10} + \cdots - 1 \)
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: | $[6, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2259436291848000000000\)
\(\medspace = 2^{12}\cdot 3^{24}\cdot 5^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.36\) | 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: | $2\cdot 3^{4/3}5^{1/2}\approx 19.349808478363364$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}-\frac{2}{5}a^{11}-\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{13}-\frac{2}{5}a^{8}-\frac{1}{5}a^{7}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{14}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{5}a^{15}-\frac{2}{5}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}$, $\frac{1}{95}a^{16}-\frac{2}{95}a^{15}+\frac{2}{95}a^{13}+\frac{7}{95}a^{12}+\frac{34}{95}a^{11}+\frac{31}{95}a^{10}-\frac{4}{95}a^{9}+\frac{42}{95}a^{8}+\frac{26}{95}a^{7}-\frac{6}{19}a^{6}-\frac{11}{95}a^{5}-\frac{33}{95}a^{4}-\frac{28}{95}a^{3}+\frac{33}{95}a^{2}-\frac{14}{95}a+\frac{46}{95}$, $\frac{1}{1519413755}a^{17}-\frac{7394643}{1519413755}a^{16}-\frac{9466821}{1519413755}a^{15}-\frac{149132176}{1519413755}a^{14}-\frac{41528887}{1519413755}a^{13}+\frac{107213891}{1519413755}a^{12}+\frac{1524261}{3383995}a^{11}-\frac{285350033}{1519413755}a^{10}-\frac{78261727}{1519413755}a^{9}-\frac{19149656}{303882751}a^{8}+\frac{25003882}{79969145}a^{7}-\frac{70798287}{1519413755}a^{6}-\frac{26278124}{303882751}a^{5}-\frac{2963992}{8488345}a^{4}+\frac{121277203}{303882751}a^{3}+\frac{197376302}{1519413755}a^{2}+\frac{196061727}{1519413755}a-\frac{688023346}{1519413755}$
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 $1$
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: |
$\frac{4222683}{8488345}a^{17}-\frac{23453349}{8488345}a^{16}+\frac{10773565}{1697669}a^{15}-\frac{31014309}{8488345}a^{14}-\frac{171242124}{8488345}a^{13}+\frac{572325528}{8488345}a^{12}-\frac{2012187}{18905}a^{11}+\frac{7482993}{89351}a^{10}+\frac{258117323}{8488345}a^{9}-\frac{1621748478}{8488345}a^{8}+\frac{2642417379}{8488345}a^{7}-\frac{2893930678}{8488345}a^{6}+\frac{2168067801}{8488345}a^{5}-\frac{858726669}{8488345}a^{4}+\frac{9573743}{8488345}a^{3}+\frac{93301107}{8488345}a^{2}-\frac{3520803}{8488345}a+\frac{7103231}{8488345}$, $\frac{1155795243}{1519413755}a^{17}-\frac{352478243}{79969145}a^{16}+\frac{15816010774}{1519413755}a^{15}-\frac{10021131009}{1519413755}a^{14}-\frac{9614772265}{303882751}a^{13}+\frac{33284234222}{303882751}a^{12}-\frac{593590219}{3383995}a^{11}+\frac{214501841088}{1519413755}a^{10}+\frac{66611768178}{1519413755}a^{9}-\frac{470044427701}{1519413755}a^{8}+\frac{773724994636}{1519413755}a^{7}-\frac{171102314172}{303882751}a^{6}+\frac{654997908834}{1519413755}a^{5}-\frac{1471449282}{8488345}a^{4}+\frac{9904640809}{1519413755}a^{3}+\frac{20207304576}{1519413755}a^{2}-\frac{3946737999}{1519413755}a+\frac{1955710076}{1519413755}$, $\frac{568069924}{1519413755}a^{17}-\frac{119727119}{79969145}a^{16}+\frac{2820116587}{1519413755}a^{15}+\frac{4574678414}{1519413755}a^{14}-\frac{24297220148}{1519413755}a^{13}+\frac{39772316039}{1519413755}a^{12}-\frac{50126699}{3383995}a^{11}-\frac{7018081201}{303882751}a^{10}+\frac{20672183144}{303882751}a^{9}-\frac{117373388539}{1519413755}a^{8}+\frac{13733111757}{303882751}a^{7}-\frac{4664085548}{1519413755}a^{6}-\frac{81033920627}{1519413755}a^{5}+\frac{594723819}{8488345}a^{4}-\frac{28622931163}{1519413755}a^{3}-\frac{15048125443}{1519413755}a^{2}+\frac{8499263159}{1519413755}a+\frac{31373481}{1519413755}$, $\frac{236667436}{303882751}a^{17}-\frac{62710694}{15993829}a^{16}+\frac{11829105047}{1519413755}a^{15}-\frac{1848172533}{1519413755}a^{14}-\frac{49888763347}{1519413755}a^{13}+\frac{133369018021}{1519413755}a^{12}-\frac{393877358}{3383995}a^{11}+\frac{18932985192}{303882751}a^{10}+\frac{131257997676}{1519413755}a^{9}-\frac{380724783877}{1519413755}a^{8}+\frac{515090494031}{1519413755}a^{7}-\frac{505829451147}{1519413755}a^{6}+\frac{313905088444}{1519413755}a^{5}-\frac{343170751}{8488345}a^{4}-\frac{17592162336}{1519413755}a^{3}-\frac{6444751798}{1519413755}a^{2}+\frac{973624470}{303882751}a+\frac{2134059582}{1519413755}$, $\frac{68723821}{303882751}a^{17}-\frac{375177022}{303882751}a^{16}+\frac{4337088624}{1519413755}a^{15}-\frac{2767260223}{1519413755}a^{14}-\frac{12875701727}{1519413755}a^{13}+\frac{45123328849}{1519413755}a^{12}-\frac{166555537}{3383995}a^{11}+\frac{65700277458}{1519413755}a^{10}+\frac{1466582704}{303882751}a^{9}-\frac{121024888454}{1519413755}a^{8}+\frac{219121228452}{1519413755}a^{7}-\frac{258595149833}{1519413755}a^{6}+\frac{2213105208}{15993829}a^{5}-\frac{584002851}{8488345}a^{4}+\frac{23057413352}{1519413755}a^{3}+\frac{7364474443}{1519413755}a^{2}-\frac{6946270218}{1519413755}a-\frac{105542892}{303882751}$, $\frac{943859392}{1519413755}a^{17}-\frac{5293788257}{1519413755}a^{16}+\frac{12164386259}{1519413755}a^{15}-\frac{6792436688}{1519413755}a^{14}-\frac{2070891002}{79969145}a^{13}+\frac{6834011393}{79969145}a^{12}-\frac{89893378}{676799}a^{11}+\frac{152929367903}{1519413755}a^{10}+\frac{68832149792}{1519413755}a^{9}-\frac{74422776540}{303882751}a^{8}+\frac{117572783523}{303882751}a^{7}-\frac{630237538423}{1519413755}a^{6}+\frac{462512967337}{1519413755}a^{5}-\frac{931783417}{8488345}a^{4}-\frac{12019445851}{1519413755}a^{3}+\frac{17313850099}{1519413755}a^{2}+\frac{127984454}{79969145}a+\frac{150414986}{303882751}$, $\frac{615267256}{1519413755}a^{17}-\frac{193826351}{79969145}a^{16}+\frac{1801797692}{303882751}a^{15}-\frac{6422850947}{1519413755}a^{14}-\frac{25591543199}{1519413755}a^{13}+\frac{93596701982}{1519413755}a^{12}-\frac{343750659}{3383995}a^{11}+\frac{25951066393}{303882751}a^{10}+\frac{27897081956}{1519413755}a^{9}-\frac{263351395338}{1519413755}a^{8}+\frac{446424935246}{1519413755}a^{7}-\frac{501165365599}{1519413755}a^{6}+\frac{394939009071}{1519413755}a^{5}-\frac{187578914}{1697669}a^{4}+\frac{11030768827}{1519413755}a^{3}+\frac{1720674729}{303882751}a^{2}-\frac{3631140809}{1519413755}a+\frac{2102686101}{1519413755}$, $\frac{385937328}{1519413755}a^{17}-\frac{2414200761}{1519413755}a^{16}+\frac{6192819684}{1519413755}a^{15}-\frac{5167093322}{1519413755}a^{14}-\frac{15719421344}{1519413755}a^{13}+\frac{63106878233}{1519413755}a^{12}-\frac{243612231}{3383995}a^{11}+\frac{98909082279}{1519413755}a^{10}+\frac{72540259}{15993829}a^{9}-\frac{175773691928}{1519413755}a^{8}+\frac{315580690612}{1519413755}a^{7}-\frac{364351482387}{1519413755}a^{6}+\frac{298423866326}{1519413755}a^{5}-\frac{782585404}{8488345}a^{4}+\frac{17289011824}{1519413755}a^{3}+\frac{286636282}{79969145}a^{2}-\frac{794027949}{1519413755}a+\frac{2600566206}{1519413755}$, $\frac{250163492}{1519413755}a^{17}-\frac{1301380191}{1519413755}a^{16}+\frac{2633481191}{1519413755}a^{15}-\frac{701083448}{1519413755}a^{14}-\frac{2045737935}{303882751}a^{13}+\frac{28732638013}{1519413755}a^{12}-\frac{89939446}{3383995}a^{11}+\frac{27362531207}{1519413755}a^{10}+\frac{18189837392}{1519413755}a^{9}-\frac{75063557856}{1519413755}a^{8}+\frac{23075046728}{303882751}a^{7}-\frac{130496717514}{1519413755}a^{6}+\frac{19700081490}{303882751}a^{5}-\frac{244932949}{8488345}a^{4}+\frac{20276024326}{1519413755}a^{3}-\frac{6648222162}{1519413755}a^{2}-\frac{1232425118}{1519413755}a-\frac{375682798}{1519413755}$, $\frac{677435809}{1519413755}a^{17}-\frac{3824360526}{1519413755}a^{16}+\frac{8710474833}{1519413755}a^{15}-\frac{4731554201}{1519413755}a^{14}-\frac{1488402243}{79969145}a^{13}+\frac{4862180987}{79969145}a^{12}-\frac{318139026}{3383995}a^{11}+\frac{109766441069}{1519413755}a^{10}+\frac{43241227382}{1519413755}a^{9}-\frac{254413685712}{1519413755}a^{8}+\frac{409250118312}{1519413755}a^{7}-\frac{453815688071}{1519413755}a^{6}+\frac{70023481634}{303882751}a^{5}-\frac{165012905}{1697669}a^{4}+\frac{5671962800}{303882751}a^{3}-\frac{10270810403}{1519413755}a^{2}+\frac{155301771}{79969145}a+\frac{1130988692}{1519413755}$, $\frac{1046015209}{1519413755}a^{17}-\frac{1272290328}{303882751}a^{16}+\frac{3116531750}{303882751}a^{15}-\frac{2211792690}{303882751}a^{14}-\frac{44612160319}{1519413755}a^{13}+\frac{162375460322}{1519413755}a^{12}-\frac{591748171}{3383995}a^{11}+\frac{43918220075}{303882751}a^{10}+\frac{56528251501}{1519413755}a^{9}-\frac{461872020776}{1519413755}a^{8}+\frac{153609521877}{303882751}a^{7}-\frac{851467345132}{1519413755}a^{6}+\frac{131983921186}{303882751}a^{5}-\frac{77929298}{446755}a^{4}+\frac{3555954114}{1519413755}a^{3}+\frac{20706362617}{1519413755}a^{2}-\frac{670940181}{1519413755}a+\frac{2769890892}{1519413755}$
| 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: | \( 4306.402696834701 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 4306.402696834701 \cdot 1}{2\cdot\sqrt{2259436291848000000000}}\cr\approx \mathstrut & 0.178379021437612 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*x - 1)
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 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*x - 1, 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 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*x - 1);
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 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*x - 1);
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
$C_6\times S_3$ (as 18T6):
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 36 |
| The 18 conjugacy class representatives for $S_3 \times C_6$ |
| Character table for $S_3 \times C_6$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 3.1.1620.1, 6.6.820125.1, 6.2.13122000.3, 9.3.4251528000.1 |
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
| Galois closure: | data not computed |
| Degree 12 sibling: | 12.0.419904000000.1 |
| Degree 18 sibling: | 18.0.1156831381426176000000.1 |
| Minimal sibling: | 12.0.419904000000.1 |
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 | R | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{9}$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
|
\(3\)
| Deg $18$ | $3$ | $6$ | $24$ | |||
|
\(5\)
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |