Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 9*x^16 - 57*x^15 + 143*x^14 - 3*x^13 + 1674*x^12 + 1233*x^11 + 5724*x^10 + 2007*x^9 + 10875*x^8 + 1305*x^7 + 10340*x^6 + 858*x^5 + 5358*x^4 + 381*x^3 + 1558*x^2 + 165*x + 183)
gp: K = bnfinit(y^18 - 3*y^17 + 9*y^16 - 57*y^15 + 143*y^14 - 3*y^13 + 1674*y^12 + 1233*y^11 + 5724*y^10 + 2007*y^9 + 10875*y^8 + 1305*y^7 + 10340*y^6 + 858*y^5 + 5358*y^4 + 381*y^3 + 1558*y^2 + 165*y + 183, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 + 9*x^16 - 57*x^15 + 143*x^14 - 3*x^13 + 1674*x^12 + 1233*x^11 + 5724*x^10 + 2007*x^9 + 10875*x^8 + 1305*x^7 + 10340*x^6 + 858*x^5 + 5358*x^4 + 381*x^3 + 1558*x^2 + 165*x + 183);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 + 9*x^16 - 57*x^15 + 143*x^14 - 3*x^13 + 1674*x^12 + 1233*x^11 + 5724*x^10 + 2007*x^9 + 10875*x^8 + 1305*x^7 + 10340*x^6 + 858*x^5 + 5358*x^4 + 381*x^3 + 1558*x^2 + 165*x + 183)
\( x^{18} - 3 x^{17} + 9 x^{16} - 57 x^{15} + 143 x^{14} - 3 x^{13} + 1674 x^{12} + 1233 x^{11} + \cdots + 183 \)
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: |
\(-204985922317940382893914042368\)
\(\medspace = -\,2^{12}\cdot 3^{9}\cdot 7^{6}\cdot 43^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(42.50\) | 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: | $2^{2/3}3^{1/2}7^{2/3}43^{5/6}\approx 231.1371199229907$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(43\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{6}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{7}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{8}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{9}+\frac{1}{3}a^{5}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{3}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{14}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{5}-\frac{4}{9}a^{4}-\frac{1}{9}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{36\!\cdots\!43}a^{17}-\frac{13\!\cdots\!53}{36\!\cdots\!43}a^{16}+\frac{20\!\cdots\!69}{36\!\cdots\!43}a^{15}+\frac{31\!\cdots\!10}{36\!\cdots\!43}a^{14}+\frac{10\!\cdots\!71}{12\!\cdots\!81}a^{13}-\frac{12\!\cdots\!04}{12\!\cdots\!81}a^{12}+\frac{17\!\cdots\!77}{40\!\cdots\!27}a^{11}-\frac{12\!\cdots\!11}{12\!\cdots\!81}a^{10}-\frac{39\!\cdots\!30}{12\!\cdots\!81}a^{9}+\frac{12\!\cdots\!85}{40\!\cdots\!27}a^{8}-\frac{12\!\cdots\!46}{40\!\cdots\!27}a^{7}+\frac{53\!\cdots\!33}{12\!\cdots\!81}a^{6}+\frac{14\!\cdots\!42}{36\!\cdots\!43}a^{5}+\frac{57\!\cdots\!32}{36\!\cdots\!43}a^{4}-\frac{83\!\cdots\!75}{36\!\cdots\!43}a^{3}-\frac{11\!\cdots\!63}{36\!\cdots\!43}a^{2}+\frac{16\!\cdots\!69}{40\!\cdots\!27}a-\frac{32\!\cdots\!19}{12\!\cdots\!81}$
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
$C_{3}$, which has order $3$ (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: |
\( -\frac{149131692413739326}{37703520132935729883} a^{17} + \frac{473976989037599438}{37703520132935729883} a^{16} - \frac{1461017701596462538}{37703520132935729883} a^{15} + \frac{2960482522889615091}{12567840044311909961} a^{14} - \frac{7736448795665348035}{12567840044311909961} a^{13} + \frac{2169783114008740096}{12567840044311909961} a^{12} - \frac{85392600558717187811}{12567840044311909961} a^{11} - \frac{46351121725856008764}{12567840044311909961} a^{10} - \frac{292530515696235301892}{12567840044311909961} a^{9} - \frac{51723144029576407028}{12567840044311909961} a^{8} - \frac{557309903511451615850}{12567840044311909961} a^{7} + \frac{66395116999951970188}{12567840044311909961} a^{6} - \frac{1629414233231806567579}{37703520132935729883} a^{5} + \frac{355354514741260369567}{37703520132935729883} a^{4} - \frac{721259331143792675231}{37703520132935729883} a^{3} + \frac{56479911217287596864}{12567840044311909961} a^{2} - \frac{45730901114817469945}{12567840044311909961} a + \frac{17103887341629606118}{12567840044311909961} \)
(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: |
$\frac{35\!\cdots\!26}{36\!\cdots\!43}a^{17}-\frac{10\!\cdots\!87}{36\!\cdots\!43}a^{16}+\frac{30\!\cdots\!95}{36\!\cdots\!43}a^{15}-\frac{19\!\cdots\!06}{36\!\cdots\!43}a^{14}+\frac{55\!\cdots\!13}{40\!\cdots\!27}a^{13}+\frac{17\!\cdots\!27}{12\!\cdots\!81}a^{12}+\frac{19\!\cdots\!38}{12\!\cdots\!81}a^{11}+\frac{15\!\cdots\!53}{12\!\cdots\!81}a^{10}+\frac{21\!\cdots\!82}{40\!\cdots\!27}a^{9}+\frac{73\!\cdots\!37}{40\!\cdots\!27}a^{8}+\frac{11\!\cdots\!93}{12\!\cdots\!81}a^{7}+\frac{77\!\cdots\!79}{12\!\cdots\!81}a^{6}+\frac{24\!\cdots\!65}{36\!\cdots\!43}a^{5}+\frac{89\!\cdots\!99}{36\!\cdots\!43}a^{4}+\frac{73\!\cdots\!39}{36\!\cdots\!43}a^{3}-\frac{71\!\cdots\!73}{36\!\cdots\!43}a^{2}+\frac{91\!\cdots\!70}{40\!\cdots\!27}a-\frac{40\!\cdots\!97}{12\!\cdots\!81}$, $\frac{11\!\cdots\!88}{12\!\cdots\!81}a^{17}-\frac{26\!\cdots\!09}{12\!\cdots\!81}a^{16}+\frac{79\!\cdots\!27}{12\!\cdots\!81}a^{15}-\frac{57\!\cdots\!14}{12\!\cdots\!81}a^{14}+\frac{12\!\cdots\!59}{12\!\cdots\!81}a^{13}+\frac{87\!\cdots\!06}{12\!\cdots\!81}a^{12}+\frac{18\!\cdots\!85}{12\!\cdots\!81}a^{11}+\frac{24\!\cdots\!03}{12\!\cdots\!81}a^{10}+\frac{71\!\cdots\!14}{12\!\cdots\!81}a^{9}+\frac{52\!\cdots\!37}{12\!\cdots\!81}a^{8}+\frac{12\!\cdots\!24}{12\!\cdots\!81}a^{7}+\frac{58\!\cdots\!97}{12\!\cdots\!81}a^{6}+\frac{34\!\cdots\!97}{40\!\cdots\!27}a^{5}+\frac{29\!\cdots\!86}{12\!\cdots\!81}a^{4}+\frac{42\!\cdots\!71}{12\!\cdots\!81}a^{3}+\frac{26\!\cdots\!61}{12\!\cdots\!81}a^{2}+\frac{17\!\cdots\!37}{40\!\cdots\!27}a-\frac{62\!\cdots\!18}{40\!\cdots\!27}$, $\frac{31\!\cdots\!46}{36\!\cdots\!43}a^{17}-\frac{36\!\cdots\!59}{12\!\cdots\!81}a^{16}+\frac{32\!\cdots\!70}{36\!\cdots\!43}a^{15}-\frac{63\!\cdots\!87}{12\!\cdots\!81}a^{14}+\frac{17\!\cdots\!36}{12\!\cdots\!81}a^{13}-\frac{22\!\cdots\!31}{40\!\cdots\!27}a^{12}+\frac{17\!\cdots\!25}{12\!\cdots\!81}a^{11}+\frac{11\!\cdots\!02}{40\!\cdots\!27}a^{10}+\frac{48\!\cdots\!79}{12\!\cdots\!81}a^{9}-\frac{50\!\cdots\!36}{40\!\cdots\!27}a^{8}+\frac{85\!\cdots\!15}{12\!\cdots\!81}a^{7}-\frac{18\!\cdots\!72}{40\!\cdots\!27}a^{6}+\frac{20\!\cdots\!11}{36\!\cdots\!43}a^{5}-\frac{58\!\cdots\!48}{12\!\cdots\!81}a^{4}+\frac{67\!\cdots\!75}{36\!\cdots\!43}a^{3}-\frac{30\!\cdots\!50}{12\!\cdots\!81}a^{2}+\frac{15\!\cdots\!16}{12\!\cdots\!81}a-\frac{18\!\cdots\!58}{40\!\cdots\!27}$, $\frac{12\!\cdots\!73}{36\!\cdots\!43}a^{17}-\frac{42\!\cdots\!91}{36\!\cdots\!43}a^{16}+\frac{12\!\cdots\!36}{36\!\cdots\!43}a^{15}-\frac{73\!\cdots\!13}{36\!\cdots\!43}a^{14}+\frac{68\!\cdots\!46}{12\!\cdots\!81}a^{13}-\frac{25\!\cdots\!91}{12\!\cdots\!81}a^{12}+\frac{65\!\cdots\!02}{12\!\cdots\!81}a^{11}+\frac{16\!\cdots\!84}{12\!\cdots\!81}a^{10}+\frac{19\!\cdots\!54}{12\!\cdots\!81}a^{9}-\frac{27\!\cdots\!16}{12\!\cdots\!81}a^{8}+\frac{35\!\cdots\!01}{12\!\cdots\!81}a^{7}-\frac{12\!\cdots\!83}{12\!\cdots\!81}a^{6}+\frac{91\!\cdots\!94}{36\!\cdots\!43}a^{5}-\frac{20\!\cdots\!34}{36\!\cdots\!43}a^{4}+\frac{29\!\cdots\!20}{36\!\cdots\!43}a^{3}-\frac{12\!\cdots\!39}{36\!\cdots\!43}a^{2}+\frac{51\!\cdots\!00}{40\!\cdots\!27}a-\frac{28\!\cdots\!14}{12\!\cdots\!81}$, $\frac{10\!\cdots\!76}{36\!\cdots\!43}a^{17}-\frac{41\!\cdots\!27}{12\!\cdots\!81}a^{16}+\frac{31\!\cdots\!93}{36\!\cdots\!43}a^{15}-\frac{40\!\cdots\!70}{12\!\cdots\!81}a^{14}+\frac{69\!\cdots\!35}{40\!\cdots\!27}a^{13}-\frac{11\!\cdots\!95}{40\!\cdots\!27}a^{12}+\frac{85\!\cdots\!40}{40\!\cdots\!27}a^{11}-\frac{48\!\cdots\!52}{12\!\cdots\!81}a^{10}-\frac{51\!\cdots\!69}{12\!\cdots\!81}a^{9}-\frac{61\!\cdots\!40}{40\!\cdots\!27}a^{8}-\frac{36\!\cdots\!44}{40\!\cdots\!27}a^{7}-\frac{31\!\cdots\!69}{12\!\cdots\!81}a^{6}-\frac{40\!\cdots\!43}{36\!\cdots\!43}a^{5}-\frac{23\!\cdots\!16}{12\!\cdots\!81}a^{4}-\frac{24\!\cdots\!78}{36\!\cdots\!43}a^{3}-\frac{75\!\cdots\!33}{12\!\cdots\!81}a^{2}-\frac{13\!\cdots\!21}{12\!\cdots\!81}a-\frac{10\!\cdots\!81}{40\!\cdots\!27}$, $\frac{90\!\cdots\!22}{36\!\cdots\!43}a^{17}-\frac{83\!\cdots\!63}{12\!\cdots\!81}a^{16}+\frac{74\!\cdots\!55}{36\!\cdots\!43}a^{15}-\frac{16\!\cdots\!53}{12\!\cdots\!81}a^{14}+\frac{38\!\cdots\!74}{12\!\cdots\!81}a^{13}+\frac{10\!\cdots\!62}{12\!\cdots\!81}a^{12}+\frac{16\!\cdots\!26}{40\!\cdots\!27}a^{11}+\frac{16\!\cdots\!09}{40\!\cdots\!27}a^{10}+\frac{17\!\cdots\!54}{12\!\cdots\!81}a^{9}+\frac{10\!\cdots\!37}{12\!\cdots\!81}a^{8}+\frac{10\!\cdots\!49}{40\!\cdots\!27}a^{7}+\frac{38\!\cdots\!61}{40\!\cdots\!27}a^{6}+\frac{84\!\cdots\!41}{36\!\cdots\!43}a^{5}+\frac{11\!\cdots\!66}{12\!\cdots\!81}a^{4}+\frac{34\!\cdots\!29}{36\!\cdots\!43}a^{3}+\frac{59\!\cdots\!49}{12\!\cdots\!81}a^{2}+\frac{21\!\cdots\!01}{12\!\cdots\!81}a+\frac{62\!\cdots\!37}{40\!\cdots\!27}$, $\frac{88\!\cdots\!52}{36\!\cdots\!43}a^{17}-\frac{29\!\cdots\!14}{36\!\cdots\!43}a^{16}+\frac{86\!\cdots\!81}{36\!\cdots\!43}a^{15}-\frac{52\!\cdots\!01}{36\!\cdots\!43}a^{14}+\frac{47\!\cdots\!98}{12\!\cdots\!81}a^{13}-\frac{10\!\cdots\!48}{12\!\cdots\!81}a^{12}+\frac{47\!\cdots\!52}{12\!\cdots\!81}a^{11}+\frac{56\!\cdots\!58}{40\!\cdots\!27}a^{10}+\frac{13\!\cdots\!24}{12\!\cdots\!81}a^{9}-\frac{78\!\cdots\!64}{40\!\cdots\!27}a^{8}+\frac{23\!\cdots\!52}{12\!\cdots\!81}a^{7}-\frac{34\!\cdots\!01}{40\!\cdots\!27}a^{6}+\frac{57\!\cdots\!84}{36\!\cdots\!43}a^{5}-\frac{26\!\cdots\!09}{36\!\cdots\!43}a^{4}+\frac{20\!\cdots\!87}{36\!\cdots\!43}a^{3}-\frac{65\!\cdots\!98}{36\!\cdots\!43}a^{2}+\frac{24\!\cdots\!05}{40\!\cdots\!27}a-\frac{37\!\cdots\!17}{12\!\cdots\!81}$, $\frac{29\!\cdots\!31}{36\!\cdots\!43}a^{17}+\frac{31\!\cdots\!17}{36\!\cdots\!43}a^{16}-\frac{85\!\cdots\!39}{36\!\cdots\!43}a^{15}+\frac{13\!\cdots\!22}{36\!\cdots\!43}a^{14}-\frac{19\!\cdots\!49}{40\!\cdots\!27}a^{13}+\frac{16\!\cdots\!89}{12\!\cdots\!81}a^{12}+\frac{86\!\cdots\!07}{40\!\cdots\!27}a^{11}+\frac{22\!\cdots\!64}{12\!\cdots\!81}a^{10}+\frac{87\!\cdots\!38}{40\!\cdots\!27}a^{9}+\frac{65\!\cdots\!32}{12\!\cdots\!81}a^{8}+\frac{11\!\cdots\!79}{40\!\cdots\!27}a^{7}+\frac{96\!\cdots\!19}{12\!\cdots\!81}a^{6}+\frac{79\!\cdots\!24}{36\!\cdots\!43}a^{5}+\frac{17\!\cdots\!19}{36\!\cdots\!43}a^{4}+\frac{31\!\cdots\!41}{36\!\cdots\!43}a^{3}+\frac{64\!\cdots\!50}{36\!\cdots\!43}a^{2}+\frac{11\!\cdots\!42}{40\!\cdots\!27}a+\frac{27\!\cdots\!25}{12\!\cdots\!81}$
| 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: | \( 55890127.22994664 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 55890127.22994664 \cdot 3}{6\cdot\sqrt{204985922317940382893914042368}}\cr\approx \mathstrut & 0.942023852880126 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 9*x^16 - 57*x^15 + 143*x^14 - 3*x^13 + 1674*x^12 + 1233*x^11 + 5724*x^10 + 2007*x^9 + 10875*x^8 + 1305*x^7 + 10340*x^6 + 858*x^5 + 5358*x^4 + 381*x^3 + 1558*x^2 + 165*x + 183)
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 - 3*x^17 + 9*x^16 - 57*x^15 + 143*x^14 - 3*x^13 + 1674*x^12 + 1233*x^11 + 5724*x^10 + 2007*x^9 + 10875*x^8 + 1305*x^7 + 10340*x^6 + 858*x^5 + 5358*x^4 + 381*x^3 + 1558*x^2 + 165*x + 183, 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 - 3*x^17 + 9*x^16 - 57*x^15 + 143*x^14 - 3*x^13 + 1674*x^12 + 1233*x^11 + 5724*x^10 + 2007*x^9 + 10875*x^8 + 1305*x^7 + 10340*x^6 + 858*x^5 + 5358*x^4 + 381*x^3 + 1558*x^2 + 165*x + 183);
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 - 3*x^17 + 9*x^16 - 57*x^15 + 143*x^14 - 3*x^13 + 1674*x^12 + 1233*x^11 + 5724*x^10 + 2007*x^9 + 10875*x^8 + 1305*x^7 + 10340*x^6 + 858*x^5 + 5358*x^4 + 381*x^3 + 1558*x^2 + 165*x + 183);
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
$C_3\times S_3^2$ (as 18T46):
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 108 |
| The 27 conjugacy class representatives for $C_3\times S_3^2$ |
| Character table for $C_3\times S_3^2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.172.1, 6.0.2446227.1, 6.0.798768.3 |
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 12 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.5237298151345428666624.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.12.1 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
|
\(3\)
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(7\)
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 7.3.2.1 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.6.4.2 | $x^{6} - 42 x^{3} + 147$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 7.6.0.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(43\)
| 43.3.2.2 | $x^{3} + 301$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 43.3.0.1 | $x^{3} + x + 40$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 43.6.3.2 | $x^{6} + 131 x^{4} + 80 x^{3} + 5548 x^{2} - 10240 x + 77452$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 43.6.5.4 | $x^{6} + 387$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |