Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 2*x^20 + 5*x^18 - x^17 + 6*x^16 - 3*x^15 + 9*x^14 - x^13 + 7*x^12 - 3*x^11 + 13*x^10 + 5*x^8 + 3*x^7 + 10*x^6 + x^4 + 4*x^2 + 2*x + 1)
gp: K = bnfinit(y^22 + 2*y^20 + 5*y^18 - y^17 + 6*y^16 - 3*y^15 + 9*y^14 - y^13 + 7*y^12 - 3*y^11 + 13*y^10 + 5*y^8 + 3*y^7 + 10*y^6 + y^4 + 4*y^2 + 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 2*x^20 + 5*x^18 - x^17 + 6*x^16 - 3*x^15 + 9*x^14 - x^13 + 7*x^12 - 3*x^11 + 13*x^10 + 5*x^8 + 3*x^7 + 10*x^6 + x^4 + 4*x^2 + 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 2*x^20 + 5*x^18 - x^17 + 6*x^16 - 3*x^15 + 9*x^14 - x^13 + 7*x^12 - 3*x^11 + 13*x^10 + 5*x^8 + 3*x^7 + 10*x^6 + x^4 + 4*x^2 + 2*x + 1)
\( x^{22} + 2 x^{20} + 5 x^{18} - x^{17} + 6 x^{16} - 3 x^{15} + 9 x^{14} - x^{13} + 7 x^{12} - 3 x^{11} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-178400853291024459894627\)
\(\medspace = -\,3^{11}\cdot 1003532779^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.40\) | 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: | $3^{1/2}1003532779^{1/2}\approx 54868.91959023797$ | ||
| Ramified primes: |
\(3\), \(1003532779\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{15}-\frac{1}{3}a^{14}+\frac{1}{3}a^{12}-\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{17}+\frac{1}{3}a^{15}+\frac{1}{3}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{18}-\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{19}-\frac{1}{3}a^{15}-\frac{1}{3}a^{14}+\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{20}+\frac{1}{3}a^{13}+\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{5397333}a^{21}+\frac{441266}{5397333}a^{20}-\frac{301661}{5397333}a^{19}-\frac{39386}{1799111}a^{18}-\frac{871243}{5397333}a^{17}+\frac{316840}{5397333}a^{16}-\frac{1794586}{5397333}a^{15}+\frac{1514548}{5397333}a^{14}-\frac{823615}{5397333}a^{13}+\frac{506099}{1799111}a^{12}+\frac{1898719}{5397333}a^{11}-\frac{152372}{1799111}a^{10}+\frac{393048}{1799111}a^{9}+\frac{820146}{1799111}a^{8}-\frac{1880657}{5397333}a^{7}-\frac{1257233}{5397333}a^{6}-\frac{309008}{5397333}a^{5}-\frac{101438}{5397333}a^{4}+\frac{2540284}{5397333}a^{3}-\frac{2345450}{5397333}a^{2}+\frac{1847941}{5397333}a-\frac{933665}{5397333}$
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: | $10$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{2810593}{5397333} a^{21} - \frac{1635334}{5397333} a^{20} + \frac{2117703}{1799111} a^{19} - \frac{3344648}{5397333} a^{18} + \frac{5175742}{1799111} a^{17} - \frac{10279327}{5397333} a^{16} + \frac{21606998}{5397333} a^{15} - \frac{17570141}{5397333} a^{14} + \frac{33587785}{5397333} a^{13} - \frac{5417000}{1799111} a^{12} + \frac{24871499}{5397333} a^{11} - \frac{16467355}{5397333} a^{10} + \frac{41454842}{5397333} a^{9} - \frac{5366617}{1799111} a^{8} + \frac{19521955}{5397333} a^{7} + \frac{4275268}{5397333} a^{6} + \frac{7771465}{1799111} a^{5} - \frac{3009008}{5397333} a^{4} + \frac{4389352}{5397333} a^{3} + \frac{1071251}{5397333} a^{2} + \frac{4088518}{1799111} a + \frac{1934567}{1799111} \)
(order $6$)
| 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: |
$\frac{135456}{1799111}a^{21}-\frac{2810593}{5397333}a^{20}+\frac{2448070}{5397333}a^{19}-\frac{2117703}{1799111}a^{18}+\frac{5376488}{5397333}a^{17}-\frac{5311198}{1799111}a^{16}+\frac{12717535}{5397333}a^{15}-\frac{22826102}{5397333}a^{14}+\frac{21227453}{5397333}a^{13}-\frac{33994153}{5397333}a^{12}+\frac{6365192}{1799111}a^{11}-\frac{26090603}{5397333}a^{10}+\frac{21750139}{5397333}a^{9}-\frac{41454842}{5397333}a^{8}+\frac{6043897}{1799111}a^{7}-\frac{18302851}{5397333}a^{6}-\frac{211588}{5397333}a^{5}-\frac{7771465}{1799111}a^{4}+\frac{3415376}{5397333}a^{3}-\frac{4389352}{5397333}a^{2}+\frac{554221}{5397333}a-\frac{3817606}{1799111}$, $\frac{13146215}{5397333}a^{21}+\frac{520521}{1799111}a^{20}+\frac{6881174}{1799111}a^{19}+\frac{1924477}{1799111}a^{18}+\frac{55202282}{5397333}a^{17}-\frac{351090}{1799111}a^{16}+\frac{51191879}{5397333}a^{15}-\frac{13287413}{5397333}a^{14}+\frac{81707636}{5397333}a^{13}+\frac{28405441}{5397333}a^{12}+\frac{13071900}{1799111}a^{11}-\frac{6827959}{5397333}a^{10}+\frac{44922209}{1799111}a^{9}+\frac{47069564}{5397333}a^{8}-\frac{2788421}{1799111}a^{7}+\frac{75481936}{5397333}a^{6}+\frac{116028835}{5397333}a^{5}+\frac{503584}{5397333}a^{4}-\frac{29132048}{5397333}a^{3}+\frac{29222654}{5397333}a^{2}+\frac{16084401}{1799111}a+\frac{29290429}{5397333}$, $\frac{2914040}{5397333}a^{21}-\frac{4633946}{5397333}a^{20}+\frac{7807048}{5397333}a^{19}-\frac{8672473}{5397333}a^{18}+\frac{5439087}{1799111}a^{17}-\frac{7643274}{1799111}a^{16}+\frac{27333146}{5397333}a^{15}-\frac{32097508}{5397333}a^{14}+\frac{13594317}{1799111}a^{13}-\frac{12320291}{1799111}a^{12}+\frac{9103563}{1799111}a^{11}-\frac{23109127}{5397333}a^{10}+\frac{14745544}{1799111}a^{9}-\frac{49184566}{5397333}a^{8}+\frac{28859927}{5397333}a^{7}+\frac{3228196}{5397333}a^{6}+\frac{193208}{1799111}a^{5}-\frac{7347184}{1799111}a^{4}+\frac{16392973}{5397333}a^{3}-\frac{4183661}{5397333}a^{2}+\frac{1494144}{1799111}a-\frac{961963}{5397333}$, $\frac{9092608}{5397333}a^{21}+\frac{1949965}{5397333}a^{20}+\frac{4740386}{1799111}a^{19}+\frac{5342473}{5397333}a^{18}+\frac{37452341}{5397333}a^{17}+\frac{3465530}{5397333}a^{16}+\frac{32565739}{5397333}a^{15}-\frac{1645737}{1799111}a^{14}+\frac{50570243}{5397333}a^{13}+\frac{27878062}{5397333}a^{12}+\frac{7407051}{1799111}a^{11}+\frac{532704}{1799111}a^{10}+\frac{84369250}{5397333}a^{9}+\frac{45462181}{5397333}a^{8}-\frac{2397031}{1799111}a^{7}+\frac{17748704}{1799111}a^{6}+\frac{26129573}{1799111}a^{5}+\frac{12466955}{5397333}a^{4}-\frac{25979498}{5397333}a^{3}+\frac{13953313}{5397333}a^{2}+\frac{34422725}{5397333}a+\frac{7474608}{1799111}$, $\frac{3310174}{5397333}a^{21}-\frac{395729}{5397333}a^{20}+\frac{4781983}{5397333}a^{19}+\frac{1392797}{5397333}a^{18}+\frac{12604940}{5397333}a^{17}-\frac{541541}{1799111}a^{16}+\frac{11732941}{5397333}a^{15}-\frac{497749}{1799111}a^{14}+\frac{18872615}{5397333}a^{13}+\frac{7076956}{5397333}a^{12}+\frac{1911459}{1799111}a^{11}+\frac{1611922}{5397333}a^{10}+\frac{12159592}{1799111}a^{9}+\frac{3149428}{5397333}a^{8}-\frac{3629230}{5397333}a^{7}+\frac{27384659}{5397333}a^{6}+\frac{26905435}{5397333}a^{5}-\frac{14741615}{5397333}a^{4}-\frac{94378}{5397333}a^{3}+\frac{4204219}{1799111}a^{2}+\frac{1821097}{1799111}a+\frac{675732}{1799111}$, $\frac{1516616}{5397333}a^{21}-\frac{1344345}{1799111}a^{20}+\frac{1543597}{1799111}a^{19}-\frac{7058617}{5397333}a^{18}+\frac{2934606}{1799111}a^{17}-\frac{5978220}{1799111}a^{16}+\frac{5690015}{1799111}a^{15}-\frac{7280783}{1799111}a^{14}+\frac{24076015}{5397333}a^{13}-\frac{26637058}{5397333}a^{12}+\frac{11927857}{5397333}a^{11}-\frac{10801382}{5397333}a^{10}+\frac{19309369}{5397333}a^{9}-\frac{12022578}{1799111}a^{8}+\frac{3771564}{1799111}a^{7}+\frac{10328491}{5397333}a^{6}-\frac{4681216}{1799111}a^{5}-\frac{16704197}{5397333}a^{4}+\frac{13666211}{5397333}a^{3}+\frac{3097781}{5397333}a^{2}-\frac{2245924}{5397333}a-\frac{2973980}{5397333}$, $\frac{1649137}{1799111}a^{21}-\frac{8779735}{5397333}a^{20}+\frac{4459830}{1799111}a^{19}-\frac{16239262}{5397333}a^{18}+\frac{29874142}{5397333}a^{17}-\frac{45704170}{5397333}a^{16}+\frac{17623607}{1799111}a^{15}-\frac{21250001}{1799111}a^{14}+\frac{82374466}{5397333}a^{13}-\frac{27577228}{1799111}a^{12}+\frac{61194208}{5397333}a^{11}-\frac{61797898}{5397333}a^{10}+\frac{97976017}{5397333}a^{9}-\frac{111138641}{5397333}a^{8}+\frac{55841230}{5397333}a^{7}-\frac{3502795}{5397333}a^{6}+\frac{17587739}{5397333}a^{5}-\frac{63628897}{5397333}a^{4}+\frac{40163119}{5397333}a^{3}-\frac{2600372}{5397333}a^{2}+\frac{6449605}{5397333}a-\frac{15994370}{5397333}$, $\frac{2905332}{1799111}a^{21}-\frac{3835979}{5397333}a^{20}+\frac{16710484}{5397333}a^{19}-\frac{6742082}{5397333}a^{18}+\frac{41554210}{5397333}a^{17}-\frac{26828362}{5397333}a^{16}+\frac{51944777}{5397333}a^{15}-\frac{15306752}{1799111}a^{14}+\frac{83318222}{5397333}a^{13}-\frac{13235967}{1799111}a^{12}+\frac{56099627}{5397333}a^{11}-\frac{48538483}{5397333}a^{10}+\frac{40178935}{1799111}a^{9}-\frac{16681773}{1799111}a^{8}+\frac{34565842}{5397333}a^{7}+\frac{14466397}{5397333}a^{6}+\frac{23339675}{1799111}a^{5}-\frac{42860404}{5397333}a^{4}+\frac{7428496}{5397333}a^{3}+\frac{85089}{1799111}a^{2}+\frac{31373185}{5397333}a+\frac{1228966}{5397333}$, $\frac{4747603}{5397333}a^{21}+\frac{4369891}{5397333}a^{20}+\frac{4848301}{5397333}a^{19}+\frac{3309138}{1799111}a^{18}+\frac{4940746}{1799111}a^{17}+\frac{19014974}{5397333}a^{16}+\frac{1518235}{5397333}a^{15}+\frac{21467185}{5397333}a^{14}+\frac{3396332}{5397333}a^{13}+\frac{17403532}{1799111}a^{12}-\frac{3826538}{1799111}a^{11}+\frac{26324132}{5397333}a^{10}+\frac{24389119}{5397333}a^{9}+\frac{71451971}{5397333}a^{8}-\frac{31521257}{5397333}a^{7}+\frac{14071549}{1799111}a^{6}+\frac{50350903}{5397333}a^{5}+\frac{27465142}{5397333}a^{4}-\frac{32073251}{5397333}a^{3}+\frac{15434614}{5397333}a^{2}+\frac{8131676}{1799111}a+\frac{6768786}{1799111}$, $\frac{6525079}{5397333}a^{21}+\frac{3662947}{5397333}a^{20}+\frac{7707550}{5397333}a^{19}+\frac{9739013}{5397333}a^{18}+\frac{22075040}{5397333}a^{17}+\frac{15196262}{5397333}a^{16}+\frac{3255803}{1799111}a^{15}+\frac{17831626}{5397333}a^{14}+\frac{14706179}{5397333}a^{13}+\frac{17341213}{1799111}a^{12}-\frac{3121535}{1799111}a^{11}+\frac{28114736}{5397333}a^{10}+\frac{13131849}{1799111}a^{9}+\frac{23990703}{1799111}a^{8}-\frac{42996104}{5397333}a^{7}+\frac{66878947}{5397333}a^{6}+\frac{53840729}{5397333}a^{5}+\frac{6055099}{1799111}a^{4}-\frac{35472985}{5397333}a^{3}+\frac{9304462}{1799111}a^{2}+\frac{16839691}{5397333}a+\frac{21529436}{5397333}$
| 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: | \( 622.138184002 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 622.138184002 \cdot 1}{6\cdot\sqrt{178400853291024459894627}}\cr\approx \mathstrut & 0.147916184305 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 2*x^20 + 5*x^18 - x^17 + 6*x^16 - 3*x^15 + 9*x^14 - x^13 + 7*x^12 - 3*x^11 + 13*x^10 + 5*x^8 + 3*x^7 + 10*x^6 + x^4 + 4*x^2 + 2*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^22 + 2*x^20 + 5*x^18 - x^17 + 6*x^16 - 3*x^15 + 9*x^14 - x^13 + 7*x^12 - 3*x^11 + 13*x^10 + 5*x^8 + 3*x^7 + 10*x^6 + x^4 + 4*x^2 + 2*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^22 + 2*x^20 + 5*x^18 - x^17 + 6*x^16 - 3*x^15 + 9*x^14 - x^13 + 7*x^12 - 3*x^11 + 13*x^10 + 5*x^8 + 3*x^7 + 10*x^6 + x^4 + 4*x^2 + 2*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^22 + 2*x^20 + 5*x^18 - x^17 + 6*x^16 - 3*x^15 + 9*x^14 - x^13 + 7*x^12 - 3*x^11 + 13*x^10 + 5*x^8 + 3*x^7 + 10*x^6 + x^4 + 4*x^2 + 2*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
$C_2\times S_{11}$ (as 22T47):
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 79833600 |
| The 112 conjugacy class representatives for $C_2\times S_{11}$ are not computed |
| Character table for $C_2\times S_{11}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 11.3.27095385033.1 |
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 22 sibling: | data not computed |
| Degree 44 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 | $22$ | R | ${\href{/padicField/5.8.0.1}{8} }^{2}{,}\,{\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.7.0.1}{7} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.9.0.1}{9} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | $18{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | $18{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.10.5.2 | $x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(1003532779\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |