Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 3036*x^10 + 16940*x^9 + 3435883*x^8 - 21476268*x^7 - 1841908992*x^6 + 11122519452*x^5 + 490521018900*x^4 - 2361270759804*x^3 - 62489273434236*x^2 + 167503033069380*x + 3005497215363897)
gp: K = bnfinit(y^12 - 4*y^11 - 3036*y^10 + 16940*y^9 + 3435883*y^8 - 21476268*y^7 - 1841908992*y^6 + 11122519452*y^5 + 490521018900*y^4 - 2361270759804*y^3 - 62489273434236*y^2 + 167503033069380*y + 3005497215363897, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 3036*x^10 + 16940*x^9 + 3435883*x^8 - 21476268*x^7 - 1841908992*x^6 + 11122519452*x^5 + 490521018900*x^4 - 2361270759804*x^3 - 62489273434236*x^2 + 167503033069380*x + 3005497215363897);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 - 3036*x^10 + 16940*x^9 + 3435883*x^8 - 21476268*x^7 - 1841908992*x^6 + 11122519452*x^5 + 490521018900*x^4 - 2361270759804*x^3 - 62489273434236*x^2 + 167503033069380*x + 3005497215363897)
\( x^{12} - 4 x^{11} - 3036 x^{10} + 16940 x^{9} + 3435883 x^{8} - 21476268 x^{7} - 1841908992 x^{6} + \cdots + 30\!\cdots\!97 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1003808920860966325428537416433732056776704\)
\(\medspace = 2^{24}\cdot 3^{10}\cdot 11^{20}\cdot 197^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(3163.28\) | 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: | $2^{43/16}3^{7/8}11^{20/11}197^{1/2}\approx 18500.07845030721$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\), \(197\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{7}+\frac{1}{3}a^{4}$, $\frac{1}{13\!\cdots\!91}a^{11}+\frac{83\!\cdots\!59}{13\!\cdots\!91}a^{10}+\frac{62\!\cdots\!66}{46\!\cdots\!97}a^{9}-\frac{12\!\cdots\!47}{13\!\cdots\!91}a^{8}+\frac{34\!\cdots\!23}{13\!\cdots\!91}a^{7}-\frac{85\!\cdots\!31}{46\!\cdots\!97}a^{6}+\frac{21\!\cdots\!88}{46\!\cdots\!97}a^{5}-\frac{24\!\cdots\!84}{46\!\cdots\!97}a^{4}-\frac{16\!\cdots\!12}{46\!\cdots\!97}a^{3}-\frac{99\!\cdots\!48}{15\!\cdots\!99}a^{2}-\frac{64\!\cdots\!16}{15\!\cdots\!99}a+\frac{73\!\cdots\!91}{15\!\cdots\!99}$
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: | $11$ | 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: |
$\frac{13\!\cdots\!89}{46\!\cdots\!97}a^{11}+\frac{32\!\cdots\!92}{46\!\cdots\!97}a^{10}-\frac{27\!\cdots\!58}{46\!\cdots\!97}a^{9}-\frac{19\!\cdots\!37}{15\!\cdots\!99}a^{8}+\frac{22\!\cdots\!73}{46\!\cdots\!97}a^{7}+\frac{36\!\cdots\!15}{46\!\cdots\!97}a^{6}-\frac{86\!\cdots\!86}{46\!\cdots\!97}a^{5}-\frac{10\!\cdots\!35}{46\!\cdots\!97}a^{4}+\frac{48\!\cdots\!62}{15\!\cdots\!99}a^{3}+\frac{49\!\cdots\!70}{15\!\cdots\!99}a^{2}-\frac{28\!\cdots\!60}{15\!\cdots\!99}a-\frac{26\!\cdots\!56}{15\!\cdots\!99}$, $\frac{35\!\cdots\!64}{13\!\cdots\!91}a^{11}-\frac{10\!\cdots\!46}{13\!\cdots\!91}a^{10}-\frac{27\!\cdots\!35}{46\!\cdots\!97}a^{9}+\frac{26\!\cdots\!55}{13\!\cdots\!91}a^{8}+\frac{57\!\cdots\!50}{13\!\cdots\!91}a^{7}-\frac{24\!\cdots\!65}{15\!\cdots\!99}a^{6}-\frac{13\!\cdots\!13}{15\!\cdots\!99}a^{5}+\frac{77\!\cdots\!46}{15\!\cdots\!99}a^{4}+\frac{10\!\cdots\!78}{46\!\cdots\!97}a^{3}-\frac{10\!\cdots\!52}{15\!\cdots\!99}a^{2}+\frac{75\!\cdots\!84}{15\!\cdots\!99}a+\frac{47\!\cdots\!83}{15\!\cdots\!99}$, $\frac{30\!\cdots\!57}{13\!\cdots\!91}a^{11}-\frac{19\!\cdots\!22}{13\!\cdots\!91}a^{10}-\frac{10\!\cdots\!70}{46\!\cdots\!97}a^{9}+\frac{35\!\cdots\!93}{13\!\cdots\!91}a^{8}-\frac{48\!\cdots\!01}{13\!\cdots\!91}a^{7}-\frac{78\!\cdots\!91}{46\!\cdots\!97}a^{6}+\frac{43\!\cdots\!86}{46\!\cdots\!97}a^{5}+\frac{22\!\cdots\!33}{46\!\cdots\!97}a^{4}-\frac{12\!\cdots\!94}{46\!\cdots\!97}a^{3}-\frac{93\!\cdots\!76}{15\!\cdots\!99}a^{2}+\frac{27\!\cdots\!02}{15\!\cdots\!99}a+\frac{41\!\cdots\!42}{15\!\cdots\!99}$, $\frac{11\!\cdots\!89}{15\!\cdots\!99}a^{11}-\frac{20\!\cdots\!76}{46\!\cdots\!97}a^{10}-\frac{67\!\cdots\!51}{46\!\cdots\!97}a^{9}+\frac{53\!\cdots\!11}{46\!\cdots\!97}a^{8}+\frac{48\!\cdots\!29}{15\!\cdots\!99}a^{7}-\frac{44\!\cdots\!50}{46\!\cdots\!97}a^{6}+\frac{23\!\cdots\!26}{46\!\cdots\!97}a^{5}+\frac{15\!\cdots\!01}{46\!\cdots\!97}a^{4}-\frac{33\!\cdots\!58}{15\!\cdots\!99}a^{3}-\frac{78\!\cdots\!36}{15\!\cdots\!99}a^{2}+\frac{36\!\cdots\!83}{15\!\cdots\!99}a+\frac{50\!\cdots\!09}{15\!\cdots\!99}$, $\frac{54\!\cdots\!64}{46\!\cdots\!97}a^{11}+\frac{13\!\cdots\!53}{46\!\cdots\!97}a^{10}-\frac{12\!\cdots\!35}{46\!\cdots\!97}a^{9}-\frac{26\!\cdots\!94}{46\!\cdots\!97}a^{8}+\frac{10\!\cdots\!00}{46\!\cdots\!97}a^{7}+\frac{18\!\cdots\!44}{46\!\cdots\!97}a^{6}-\frac{44\!\cdots\!31}{46\!\cdots\!97}a^{5}-\frac{21\!\cdots\!10}{15\!\cdots\!99}a^{4}+\frac{27\!\cdots\!98}{15\!\cdots\!99}a^{3}+\frac{33\!\cdots\!69}{15\!\cdots\!99}a^{2}-\frac{17\!\cdots\!96}{15\!\cdots\!99}a-\frac{19\!\cdots\!13}{15\!\cdots\!99}$, $\frac{29\!\cdots\!43}{13\!\cdots\!91}a^{11}-\frac{73\!\cdots\!24}{13\!\cdots\!91}a^{10}-\frac{26\!\cdots\!28}{46\!\cdots\!97}a^{9}+\frac{21\!\cdots\!20}{13\!\cdots\!91}a^{8}+\frac{72\!\cdots\!91}{13\!\cdots\!91}a^{7}-\frac{73\!\cdots\!50}{46\!\cdots\!97}a^{6}-\frac{82\!\cdots\!90}{46\!\cdots\!97}a^{5}+\frac{30\!\cdots\!71}{46\!\cdots\!97}a^{4}+\frac{71\!\cdots\!41}{46\!\cdots\!97}a^{3}-\frac{16\!\cdots\!14}{15\!\cdots\!99}a^{2}+\frac{38\!\cdots\!74}{15\!\cdots\!99}a+\frac{81\!\cdots\!79}{15\!\cdots\!99}$, $\frac{31\!\cdots\!50}{13\!\cdots\!91}a^{11}+\frac{56\!\cdots\!79}{13\!\cdots\!91}a^{10}-\frac{93\!\cdots\!06}{15\!\cdots\!99}a^{9}-\frac{12\!\cdots\!64}{13\!\cdots\!91}a^{8}+\frac{80\!\cdots\!05}{13\!\cdots\!91}a^{7}+\frac{35\!\cdots\!37}{46\!\cdots\!97}a^{6}-\frac{11\!\cdots\!37}{46\!\cdots\!97}a^{5}-\frac{45\!\cdots\!87}{15\!\cdots\!99}a^{4}+\frac{22\!\cdots\!12}{46\!\cdots\!97}a^{3}+\frac{78\!\cdots\!83}{15\!\cdots\!99}a^{2}-\frac{49\!\cdots\!98}{15\!\cdots\!99}a-\frac{48\!\cdots\!42}{15\!\cdots\!99}$, $\frac{54\!\cdots\!56}{15\!\cdots\!99}a^{11}+\frac{18\!\cdots\!90}{46\!\cdots\!97}a^{10}-\frac{43\!\cdots\!64}{46\!\cdots\!97}a^{9}-\frac{29\!\cdots\!89}{46\!\cdots\!97}a^{8}+\frac{41\!\cdots\!36}{46\!\cdots\!97}a^{7}+\frac{13\!\cdots\!83}{46\!\cdots\!97}a^{6}-\frac{18\!\cdots\!98}{46\!\cdots\!97}a^{5}-\frac{33\!\cdots\!62}{46\!\cdots\!97}a^{4}+\frac{12\!\cdots\!07}{15\!\cdots\!99}a^{3}-\frac{25\!\cdots\!78}{15\!\cdots\!99}a^{2}-\frac{90\!\cdots\!03}{15\!\cdots\!99}a+\frac{35\!\cdots\!29}{15\!\cdots\!99}$, $\frac{75\!\cdots\!59}{13\!\cdots\!91}a^{11}-\frac{68\!\cdots\!58}{13\!\cdots\!91}a^{10}-\frac{71\!\cdots\!15}{46\!\cdots\!97}a^{9}+\frac{25\!\cdots\!30}{13\!\cdots\!91}a^{8}+\frac{22\!\cdots\!64}{13\!\cdots\!91}a^{7}-\frac{10\!\cdots\!94}{46\!\cdots\!97}a^{6}-\frac{33\!\cdots\!79}{46\!\cdots\!97}a^{5}+\frac{51\!\cdots\!96}{46\!\cdots\!97}a^{4}+\frac{67\!\cdots\!50}{46\!\cdots\!97}a^{3}-\frac{35\!\cdots\!42}{15\!\cdots\!99}a^{2}-\frac{15\!\cdots\!08}{15\!\cdots\!99}a+\frac{25\!\cdots\!87}{15\!\cdots\!99}$, $\frac{13\!\cdots\!06}{13\!\cdots\!91}a^{11}+\frac{20\!\cdots\!77}{13\!\cdots\!91}a^{10}-\frac{12\!\cdots\!91}{46\!\cdots\!97}a^{9}-\frac{48\!\cdots\!97}{13\!\cdots\!91}a^{8}+\frac{38\!\cdots\!23}{13\!\cdots\!91}a^{7}+\frac{47\!\cdots\!81}{15\!\cdots\!99}a^{6}-\frac{19\!\cdots\!19}{15\!\cdots\!99}a^{5}-\frac{58\!\cdots\!27}{46\!\cdots\!97}a^{4}+\frac{11\!\cdots\!07}{46\!\cdots\!97}a^{3}+\frac{36\!\cdots\!20}{15\!\cdots\!99}a^{2}-\frac{26\!\cdots\!24}{15\!\cdots\!99}a-\frac{24\!\cdots\!15}{15\!\cdots\!99}$, $\frac{15\!\cdots\!23}{46\!\cdots\!97}a^{11}+\frac{11\!\cdots\!94}{15\!\cdots\!99}a^{10}-\frac{38\!\cdots\!83}{46\!\cdots\!97}a^{9}-\frac{75\!\cdots\!31}{46\!\cdots\!97}a^{8}+\frac{34\!\cdots\!09}{46\!\cdots\!97}a^{7}+\frac{57\!\cdots\!03}{46\!\cdots\!97}a^{6}-\frac{13\!\cdots\!61}{46\!\cdots\!97}a^{5}-\frac{19\!\cdots\!45}{46\!\cdots\!97}a^{4}+\frac{85\!\cdots\!18}{15\!\cdots\!99}a^{3}+\frac{10\!\cdots\!65}{15\!\cdots\!99}a^{2}-\frac{55\!\cdots\!57}{15\!\cdots\!99}a-\frac{59\!\cdots\!82}{15\!\cdots\!99}$
| 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: | \( 377943984207000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 377943984207000000 \cdot 1}{2\cdot\sqrt{1003808920860966325428537416433732056776704}}\cr\approx \mathstrut & 0.772559369269748 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 3036*x^10 + 16940*x^9 + 3435883*x^8 - 21476268*x^7 - 1841908992*x^6 + 11122519452*x^5 + 490521018900*x^4 - 2361270759804*x^3 - 62489273434236*x^2 + 167503033069380*x + 3005497215363897)
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^12 - 4*x^11 - 3036*x^10 + 16940*x^9 + 3435883*x^8 - 21476268*x^7 - 1841908992*x^6 + 11122519452*x^5 + 490521018900*x^4 - 2361270759804*x^3 - 62489273434236*x^2 + 167503033069380*x + 3005497215363897, 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^12 - 4*x^11 - 3036*x^10 + 16940*x^9 + 3435883*x^8 - 21476268*x^7 - 1841908992*x^6 + 11122519452*x^5 + 490521018900*x^4 - 2361270759804*x^3 - 62489273434236*x^2 + 167503033069380*x + 3005497215363897);
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^12 - 4*x^11 - 3036*x^10 + 16940*x^9 + 3435883*x^8 - 21476268*x^7 - 1841908992*x^6 + 11122519452*x^5 + 490521018900*x^4 - 2361270759804*x^3 - 62489273434236*x^2 + 167503033069380*x + 3005497215363897);
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
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 95040 |
| The 15 conjugacy class representatives for $M_{12}$ |
| Character table for $M_{12}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 |
| Minimal sibling: | 12.12.111534324540107369492059712937081339641856.1 |
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 | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.24.146 | $x^{12} + 12 x^{11} + 52 x^{10} + 64 x^{9} + 162 x^{8} + 376 x^{7} + 712 x^{6} + 496 x^{5} + 996 x^{4} + 832 x^{3} + 1520 x^{2} + 800 x + 584$ | $4$ | $3$ | $24$ | 12T60 | $[2, 2, 2, 3, 3]^{3}$ |
|
\(3\)
| 3.4.3.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
| 3.8.7.2 | $x^{8} + 6$ | $8$ | $1$ | $7$ | $QD_{16}$ | $[\ ]_{8}^{2}$ | |
|
\(11\)
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 11.11.20.8 | $x^{11} + 110 x^{10} + 1221$ | $11$ | $1$ | $20$ | $C_{11}$ | $[2]$ | |
|
\(197\)
| 197.2.0.1 | $x^{2} + 192 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 197.2.0.1 | $x^{2} + 192 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 197.4.2.1 | $x^{4} + 66970 x^{3} + 1127637879 x^{2} + 214058019190 x + 3496206875$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 197.4.2.1 | $x^{4} + 66970 x^{3} + 1127637879 x^{2} + 214058019190 x + 3496206875$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |