Normalized defining polynomial
\( x^{15} + 639 x^{13} - 6552 x^{12} + 106045 x^{11} - 1415004 x^{10} + 9994509 x^{9} - 54899712 x^{8} + \cdots + 22424000 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[9, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(-17517842124064884881429616734159236380876841984000000\)
\(\medspace = -\,2^{16}\cdot 5^{6}\cdot 17^{2}\cdot 37^{6}\cdot 2803^{4}\cdot 4003^{2}\cdot 152723^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(3040.18\) | 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: | $2^{7/3}5^{1/2}17^{1/2}37^{3/4}2803^{4/5}4003^{1/2}152723^{1/2}\approx 9874328599.1493$ | ||
Ramified primes: |
\(2\), \(5\), \(17\), \(37\), \(2803\), \(4003\), \(152723\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{40}a^{11}-\frac{1}{40}a^{9}-\frac{1}{20}a^{8}-\frac{3}{8}a^{7}+\frac{3}{20}a^{6}-\frac{11}{40}a^{5}-\frac{1}{20}a^{4}+\frac{1}{10}a^{3}-\frac{7}{20}a^{2}-\frac{1}{2}a$, $\frac{1}{40}a^{12}-\frac{1}{40}a^{10}-\frac{1}{20}a^{9}-\frac{3}{8}a^{8}+\frac{3}{20}a^{7}-\frac{11}{40}a^{6}-\frac{1}{20}a^{5}+\frac{1}{10}a^{4}-\frac{7}{20}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{200}a^{13}-\frac{1}{200}a^{11}-\frac{1}{100}a^{10}-\frac{3}{40}a^{9}-\frac{37}{100}a^{8}-\frac{91}{200}a^{7}-\frac{1}{100}a^{6}+\frac{1}{50}a^{5}+\frac{33}{100}a^{4}-\frac{3}{10}a^{3}+\frac{2}{5}a^{2}$, $\frac{1}{10\!\cdots\!00}a^{14}+\frac{61\!\cdots\!96}{13\!\cdots\!75}a^{13}-\frac{86\!\cdots\!91}{10\!\cdots\!00}a^{12}-\frac{33\!\cdots\!91}{54\!\cdots\!00}a^{11}+\frac{16\!\cdots\!03}{21\!\cdots\!00}a^{10}+\frac{10\!\cdots\!03}{54\!\cdots\!00}a^{9}+\frac{31\!\cdots\!39}{10\!\cdots\!00}a^{8}+\frac{16\!\cdots\!39}{54\!\cdots\!00}a^{7}-\frac{16\!\cdots\!33}{54\!\cdots\!00}a^{6}+\frac{15\!\cdots\!33}{54\!\cdots\!00}a^{5}-\frac{12\!\cdots\!61}{27\!\cdots\!50}a^{4}-\frac{11\!\cdots\!83}{54\!\cdots\!00}a^{3}-\frac{10\!\cdots\!51}{27\!\cdots\!15}a^{2}-\frac{55\!\cdots\!21}{54\!\cdots\!30}a+\frac{18\!\cdots\!82}{54\!\cdots\!03}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
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: |
$\frac{19\!\cdots\!19}{13\!\cdots\!50}a^{14}+\frac{12\!\cdots\!99}{10\!\cdots\!00}a^{13}+\frac{63\!\cdots\!48}{68\!\cdots\!75}a^{12}-\frac{47\!\cdots\!47}{54\!\cdots\!00}a^{11}+\frac{40\!\cdots\!63}{27\!\cdots\!50}a^{10}-\frac{10\!\cdots\!69}{54\!\cdots\!00}a^{9}+\frac{88\!\cdots\!63}{68\!\cdots\!75}a^{8}-\frac{38\!\cdots\!37}{54\!\cdots\!00}a^{7}+\frac{15\!\cdots\!38}{68\!\cdots\!75}a^{6}-\frac{33\!\cdots\!31}{13\!\cdots\!50}a^{5}-\frac{80\!\cdots\!39}{27\!\cdots\!15}a^{4}+\frac{17\!\cdots\!37}{27\!\cdots\!50}a^{3}-\frac{67\!\cdots\!16}{54\!\cdots\!03}a^{2}-\frac{15\!\cdots\!97}{54\!\cdots\!30}a+\frac{51\!\cdots\!71}{54\!\cdots\!03}$, $\frac{10\!\cdots\!01}{10\!\cdots\!00}a^{14}+\frac{28\!\cdots\!11}{10\!\cdots\!60}a^{13}+\frac{65\!\cdots\!59}{10\!\cdots\!00}a^{12}-\frac{12\!\cdots\!13}{27\!\cdots\!00}a^{11}+\frac{36\!\cdots\!89}{43\!\cdots\!40}a^{10}-\frac{72\!\cdots\!09}{68\!\cdots\!75}a^{9}+\frac{65\!\cdots\!09}{10\!\cdots\!00}a^{8}-\frac{82\!\cdots\!73}{27\!\cdots\!00}a^{7}+\frac{19\!\cdots\!61}{27\!\cdots\!00}a^{6}+\frac{18\!\cdots\!99}{27\!\cdots\!00}a^{5}-\frac{16\!\cdots\!81}{54\!\cdots\!00}a^{4}-\frac{31\!\cdots\!69}{27\!\cdots\!50}a^{3}+\frac{32\!\cdots\!73}{10\!\cdots\!60}a^{2}+\frac{48\!\cdots\!37}{27\!\cdots\!15}a+\frac{36\!\cdots\!39}{54\!\cdots\!03}$, $\frac{12\!\cdots\!18}{68\!\cdots\!75}a^{14}-\frac{15\!\cdots\!73}{27\!\cdots\!50}a^{13}+\frac{61\!\cdots\!21}{54\!\cdots\!00}a^{12}-\frac{16\!\cdots\!57}{13\!\cdots\!50}a^{11}+\frac{20\!\cdots\!29}{10\!\cdots\!00}a^{10}-\frac{70\!\cdots\!93}{27\!\cdots\!00}a^{9}+\frac{10\!\cdots\!11}{54\!\cdots\!00}a^{8}-\frac{28\!\cdots\!19}{27\!\cdots\!00}a^{7}+\frac{20\!\cdots\!31}{54\!\cdots\!00}a^{6}-\frac{17\!\cdots\!33}{27\!\cdots\!00}a^{5}+\frac{11\!\cdots\!11}{54\!\cdots\!00}a^{4}+\frac{60\!\cdots\!01}{54\!\cdots\!00}a^{3}-\frac{52\!\cdots\!41}{54\!\cdots\!30}a^{2}+\frac{28\!\cdots\!17}{54\!\cdots\!30}a+\frac{66\!\cdots\!89}{54\!\cdots\!03}$, $\frac{95\!\cdots\!59}{54\!\cdots\!00}a^{14}+\frac{10\!\cdots\!01}{27\!\cdots\!50}a^{13}+\frac{76\!\cdots\!07}{68\!\cdots\!75}a^{12}-\frac{61\!\cdots\!61}{68\!\cdots\!75}a^{11}+\frac{45\!\cdots\!33}{27\!\cdots\!50}a^{10}-\frac{57\!\cdots\!73}{27\!\cdots\!00}a^{9}+\frac{35\!\cdots\!13}{27\!\cdots\!00}a^{8}-\frac{18\!\cdots\!99}{27\!\cdots\!00}a^{7}+\frac{10\!\cdots\!31}{54\!\cdots\!00}a^{6}-\frac{22\!\cdots\!03}{27\!\cdots\!00}a^{5}-\frac{48\!\cdots\!39}{13\!\cdots\!75}a^{4}+\frac{16\!\cdots\!51}{54\!\cdots\!00}a^{3}+\frac{45\!\cdots\!61}{54\!\cdots\!03}a^{2}-\frac{19\!\cdots\!19}{27\!\cdots\!15}a-\frac{97\!\cdots\!57}{54\!\cdots\!03}$, $\frac{11\!\cdots\!67}{10\!\cdots\!00}a^{14}+\frac{10\!\cdots\!07}{54\!\cdots\!00}a^{13}+\frac{73\!\cdots\!03}{10\!\cdots\!00}a^{12}-\frac{15\!\cdots\!31}{27\!\cdots\!00}a^{11}+\frac{21\!\cdots\!29}{21\!\cdots\!00}a^{10}-\frac{35\!\cdots\!87}{27\!\cdots\!00}a^{9}+\frac{87\!\cdots\!93}{10\!\cdots\!00}a^{8}-\frac{57\!\cdots\!13}{13\!\cdots\!50}a^{7}+\frac{66\!\cdots\!09}{54\!\cdots\!00}a^{6}-\frac{44\!\cdots\!88}{68\!\cdots\!75}a^{5}-\frac{25\!\cdots\!41}{10\!\cdots\!60}a^{4}+\frac{11\!\cdots\!29}{54\!\cdots\!00}a^{3}+\frac{59\!\cdots\!17}{10\!\cdots\!60}a^{2}-\frac{13\!\cdots\!26}{27\!\cdots\!15}a-\frac{67\!\cdots\!47}{54\!\cdots\!03}$, $\frac{15\!\cdots\!47}{54\!\cdots\!00}a^{14}-\frac{99\!\cdots\!91}{10\!\cdots\!00}a^{13}+\frac{97\!\cdots\!23}{54\!\cdots\!00}a^{12}-\frac{10\!\cdots\!89}{54\!\cdots\!00}a^{11}+\frac{33\!\cdots\!37}{10\!\cdots\!00}a^{10}-\frac{22\!\cdots\!43}{54\!\cdots\!00}a^{9}+\frac{15\!\cdots\!93}{54\!\cdots\!00}a^{8}-\frac{89\!\cdots\!19}{54\!\cdots\!00}a^{7}+\frac{16\!\cdots\!39}{27\!\cdots\!00}a^{6}-\frac{13\!\cdots\!77}{13\!\cdots\!50}a^{5}+\frac{93\!\cdots\!09}{27\!\cdots\!50}a^{4}+\frac{23\!\cdots\!82}{13\!\cdots\!75}a^{3}-\frac{82\!\cdots\!23}{54\!\cdots\!30}a^{2}+\frac{44\!\cdots\!51}{54\!\cdots\!30}a+\frac{10\!\cdots\!47}{54\!\cdots\!03}$, $\frac{32\!\cdots\!53}{21\!\cdots\!00}a^{14}+\frac{12\!\cdots\!39}{43\!\cdots\!24}a^{13}+\frac{21\!\cdots\!27}{21\!\cdots\!00}a^{12}-\frac{87\!\cdots\!73}{10\!\cdots\!00}a^{11}+\frac{12\!\cdots\!49}{87\!\cdots\!48}a^{10}-\frac{20\!\cdots\!21}{10\!\cdots\!00}a^{9}+\frac{25\!\cdots\!57}{21\!\cdots\!00}a^{8}-\frac{66\!\cdots\!63}{10\!\cdots\!00}a^{7}+\frac{47\!\cdots\!99}{27\!\cdots\!50}a^{6}-\frac{25\!\cdots\!09}{27\!\cdots\!50}a^{5}-\frac{36\!\cdots\!09}{10\!\cdots\!60}a^{4}+\frac{16\!\cdots\!89}{54\!\cdots\!30}a^{3}+\frac{84\!\cdots\!63}{10\!\cdots\!60}a^{2}-\frac{77\!\cdots\!89}{10\!\cdots\!06}a-\frac{96\!\cdots\!59}{54\!\cdots\!03}$, $\frac{61\!\cdots\!01}{10\!\cdots\!00}a^{14}+\frac{16\!\cdots\!78}{13\!\cdots\!75}a^{13}+\frac{39\!\cdots\!09}{10\!\cdots\!00}a^{12}-\frac{19\!\cdots\!62}{68\!\cdots\!75}a^{11}+\frac{11\!\cdots\!39}{21\!\cdots\!00}a^{10}-\frac{93\!\cdots\!93}{13\!\cdots\!50}a^{9}+\frac{45\!\cdots\!99}{10\!\cdots\!00}a^{8}-\frac{59\!\cdots\!33}{27\!\cdots\!00}a^{7}+\frac{33\!\cdots\!07}{54\!\cdots\!00}a^{6}-\frac{36\!\cdots\!43}{13\!\cdots\!50}a^{5}-\frac{15\!\cdots\!41}{13\!\cdots\!75}a^{4}+\frac{54\!\cdots\!57}{54\!\cdots\!00}a^{3}+\frac{29\!\cdots\!87}{10\!\cdots\!60}a^{2}-\frac{12\!\cdots\!71}{54\!\cdots\!30}a-\frac{31\!\cdots\!37}{54\!\cdots\!03}$, $\frac{17\!\cdots\!01}{54\!\cdots\!00}a^{14}-\frac{10\!\cdots\!99}{10\!\cdots\!00}a^{13}+\frac{11\!\cdots\!09}{54\!\cdots\!00}a^{12}-\frac{12\!\cdots\!57}{54\!\cdots\!00}a^{11}+\frac{38\!\cdots\!63}{10\!\cdots\!00}a^{10}-\frac{25\!\cdots\!19}{54\!\cdots\!00}a^{9}+\frac{18\!\cdots\!39}{54\!\cdots\!00}a^{8}-\frac{10\!\cdots\!47}{54\!\cdots\!00}a^{7}+\frac{18\!\cdots\!17}{27\!\cdots\!00}a^{6}-\frac{77\!\cdots\!23}{68\!\cdots\!75}a^{5}-\frac{63\!\cdots\!06}{13\!\cdots\!75}a^{4}+\frac{27\!\cdots\!51}{13\!\cdots\!75}a^{3}-\frac{92\!\cdots\!73}{54\!\cdots\!30}a^{2}+\frac{10\!\cdots\!53}{54\!\cdots\!30}a+\frac{13\!\cdots\!33}{54\!\cdots\!03}$, $\frac{57\!\cdots\!81}{10\!\cdots\!00}a^{14}+\frac{27\!\cdots\!67}{10\!\cdots\!00}a^{13}+\frac{85\!\cdots\!29}{10\!\cdots\!00}a^{12}+\frac{82\!\cdots\!59}{54\!\cdots\!00}a^{11}-\frac{29\!\cdots\!53}{21\!\cdots\!00}a^{10}+\frac{10\!\cdots\!93}{54\!\cdots\!00}a^{9}-\frac{30\!\cdots\!01}{10\!\cdots\!00}a^{8}+\frac{76\!\cdots\!39}{54\!\cdots\!00}a^{7}-\frac{13\!\cdots\!13}{54\!\cdots\!00}a^{6}-\frac{12\!\cdots\!61}{27\!\cdots\!00}a^{5}+\frac{10\!\cdots\!85}{21\!\cdots\!12}a^{4}-\frac{14\!\cdots\!03}{54\!\cdots\!00}a^{3}-\frac{12\!\cdots\!27}{10\!\cdots\!60}a^{2}+\frac{34\!\cdots\!69}{54\!\cdots\!30}a+\frac{96\!\cdots\!43}{54\!\cdots\!03}$, $\frac{81\!\cdots\!39}{68\!\cdots\!75}a^{14}-\frac{94\!\cdots\!32}{27\!\cdots\!15}a^{13}+\frac{20\!\cdots\!79}{27\!\cdots\!00}a^{12}-\frac{54\!\cdots\!78}{68\!\cdots\!75}a^{11}+\frac{14\!\cdots\!31}{10\!\cdots\!60}a^{10}-\frac{23\!\cdots\!07}{13\!\cdots\!50}a^{9}+\frac{33\!\cdots\!29}{27\!\cdots\!00}a^{8}-\frac{94\!\cdots\!51}{13\!\cdots\!50}a^{7}+\frac{68\!\cdots\!89}{27\!\cdots\!00}a^{6}-\frac{56\!\cdots\!37}{13\!\cdots\!50}a^{5}-\frac{45\!\cdots\!27}{27\!\cdots\!50}a^{4}+\frac{20\!\cdots\!39}{27\!\cdots\!50}a^{3}-\frac{16\!\cdots\!26}{27\!\cdots\!15}a^{2}+\frac{19\!\cdots\!53}{27\!\cdots\!15}a+\frac{49\!\cdots\!21}{54\!\cdots\!03}$
| 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: | \( 12427975138800000000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{3}\cdot 12427975138800000000000 \cdot 1}{2\cdot\sqrt{17517842124064884881429616734159236380876841984000000}}\cr\approx \mathstrut & 5.96265445690405 \end{aligned}\] (assuming GRH)
Galois group
$S_5^3.S_3$ (as 15T102):
A non-solvable group of order 10368000 |
The 140 conjugacy class representatives for $S_5^3.S_3$ |
Character table for $S_5^3.S_3$ |
Intermediate fields
3.3.148.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 18 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 36 siblings: | data not computed |
Degree 45 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 | R | ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | R | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }$ | R | $15$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\)
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.6.10.7 | $x^{6} + 2 x^{5} + 4 x^{3} + 2$ | $6$ | $1$ | $10$ | $S_4\times C_2$ | $[2, 8/3, 8/3]_{3}^{2}$ | |
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.2.2 | $x^{4} - 20 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(17\)
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.6.0.1 | $x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(37\)
| 37.4.3.1 | $x^{4} + 111$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
37.5.0.1 | $x^{5} + 10 x + 35$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
37.6.3.1 | $x^{6} + 2775 x^{5} + 2566998 x^{4} + 791680745 x^{3} + 110476893 x^{2} + 4842384300 x + 27887445532$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(2803\)
| $\Q_{2803}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2803}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{2803}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $5$ | $5$ | $1$ | $4$ | ||||
\(4003\)
| $\Q_{4003}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{4003}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
\(152723\)
| $\Q_{152723}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |