Normalized defining polynomial
\( x^{14} - 105 x^{12} - 147 x^{11} + 5271 x^{10} + 19838 x^{9} - 94150 x^{8} - 607634 x^{7} + 570164 x^{6} + 12260920 x^{5} + 42847770 x^{4} + 95169270 x^{3} + \cdots + 336238208 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-337337631352558562817384077600704\) \(\medspace = -\,2^{6}\cdot 7^{24}\cdot 31^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(210.59\) | 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: | $2^{6/7}7^{96/49}31^{1/2}\approx 456.46642613478804$ | ||
Ramified primes: | \(2\), \(7\), \(31\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-31}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{3}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{33\!\cdots\!44}a^{13}+\frac{41\!\cdots\!11}{84\!\cdots\!36}a^{12}+\frac{26\!\cdots\!67}{33\!\cdots\!44}a^{11}-\frac{14\!\cdots\!07}{33\!\cdots\!44}a^{10}-\frac{64\!\cdots\!93}{33\!\cdots\!44}a^{9}+\frac{19\!\cdots\!01}{16\!\cdots\!72}a^{8}+\frac{71\!\cdots\!89}{16\!\cdots\!72}a^{7}-\frac{31\!\cdots\!77}{16\!\cdots\!72}a^{6}-\frac{79\!\cdots\!73}{84\!\cdots\!36}a^{5}+\frac{76\!\cdots\!01}{42\!\cdots\!68}a^{4}-\frac{33\!\cdots\!71}{16\!\cdots\!72}a^{3}+\frac{68\!\cdots\!99}{16\!\cdots\!72}a^{2}+\frac{18\!\cdots\!79}{42\!\cdots\!68}a-\frac{30\!\cdots\!52}{10\!\cdots\!67}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{21}$, which has order $21$ (assuming GRH)
Unit group
Rank: | $6$ | 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: | $\frac{13\!\cdots\!81}{33\!\cdots\!44}a^{13}-\frac{16\!\cdots\!19}{84\!\cdots\!36}a^{12}-\frac{12\!\cdots\!33}{33\!\cdots\!44}a^{11}+\frac{50\!\cdots\!85}{33\!\cdots\!44}a^{10}+\frac{55\!\cdots\!03}{33\!\cdots\!44}a^{9}-\frac{44\!\cdots\!55}{16\!\cdots\!72}a^{8}-\frac{61\!\cdots\!03}{16\!\cdots\!72}a^{7}-\frac{65\!\cdots\!33}{16\!\cdots\!72}a^{6}+\frac{43\!\cdots\!99}{84\!\cdots\!36}a^{5}+\frac{75\!\cdots\!97}{42\!\cdots\!68}a^{4}+\frac{66\!\cdots\!09}{16\!\cdots\!72}a^{3}+\frac{14\!\cdots\!55}{16\!\cdots\!72}a^{2}+\frac{67\!\cdots\!61}{42\!\cdots\!68}a+\frac{14\!\cdots\!87}{10\!\cdots\!67}$, $\frac{13\!\cdots\!39}{33\!\cdots\!44}a^{13}-\frac{12\!\cdots\!97}{21\!\cdots\!34}a^{12}-\frac{13\!\cdots\!75}{33\!\cdots\!44}a^{11}-\frac{24\!\cdots\!97}{33\!\cdots\!44}a^{10}+\frac{51\!\cdots\!69}{33\!\cdots\!44}a^{9}+\frac{11\!\cdots\!33}{16\!\cdots\!72}a^{8}-\frac{32\!\cdots\!21}{16\!\cdots\!72}a^{7}-\frac{33\!\cdots\!87}{16\!\cdots\!72}a^{6}-\frac{55\!\cdots\!13}{84\!\cdots\!36}a^{5}-\frac{74\!\cdots\!55}{42\!\cdots\!68}a^{4}-\frac{73\!\cdots\!65}{16\!\cdots\!72}a^{3}-\frac{10\!\cdots\!11}{16\!\cdots\!72}a^{2}-\frac{34\!\cdots\!57}{21\!\cdots\!34}a+\frac{85\!\cdots\!81}{10\!\cdots\!67}$, $\frac{79\!\cdots\!61}{16\!\cdots\!72}a^{13}-\frac{14\!\cdots\!39}{84\!\cdots\!36}a^{12}-\frac{86\!\cdots\!25}{16\!\cdots\!72}a^{11}-\frac{73\!\cdots\!21}{16\!\cdots\!72}a^{10}+\frac{45\!\cdots\!21}{16\!\cdots\!72}a^{9}+\frac{33\!\cdots\!97}{42\!\cdots\!68}a^{8}-\frac{49\!\cdots\!45}{84\!\cdots\!36}a^{7}-\frac{22\!\cdots\!75}{84\!\cdots\!36}a^{6}+\frac{67\!\cdots\!59}{10\!\cdots\!67}a^{5}+\frac{62\!\cdots\!31}{10\!\cdots\!67}a^{4}+\frac{11\!\cdots\!49}{84\!\cdots\!36}a^{3}+\frac{20\!\cdots\!89}{84\!\cdots\!36}a^{2}+\frac{27\!\cdots\!11}{42\!\cdots\!68}a+\frac{87\!\cdots\!61}{10\!\cdots\!67}$, $\frac{11\!\cdots\!03}{33\!\cdots\!44}a^{13}-\frac{10\!\cdots\!45}{84\!\cdots\!36}a^{12}-\frac{10\!\cdots\!35}{33\!\cdots\!44}a^{11}+\frac{23\!\cdots\!79}{33\!\cdots\!44}a^{10}+\frac{53\!\cdots\!45}{33\!\cdots\!44}a^{9}+\frac{15\!\cdots\!15}{16\!\cdots\!72}a^{8}-\frac{61\!\cdots\!85}{16\!\cdots\!72}a^{7}-\frac{12\!\cdots\!47}{16\!\cdots\!72}a^{6}+\frac{41\!\cdots\!05}{84\!\cdots\!36}a^{5}+\frac{10\!\cdots\!75}{42\!\cdots\!68}a^{4}+\frac{94\!\cdots\!87}{16\!\cdots\!72}a^{3}+\frac{20\!\cdots\!25}{16\!\cdots\!72}a^{2}+\frac{10\!\cdots\!31}{42\!\cdots\!68}a+\frac{32\!\cdots\!63}{10\!\cdots\!67}$, $\frac{69\!\cdots\!27}{33\!\cdots\!44}a^{13}-\frac{37\!\cdots\!89}{42\!\cdots\!68}a^{12}-\frac{60\!\cdots\!87}{33\!\cdots\!44}a^{11}+\frac{15\!\cdots\!07}{33\!\cdots\!44}a^{10}+\frac{29\!\cdots\!61}{33\!\cdots\!44}a^{9}+\frac{46\!\cdots\!85}{16\!\cdots\!72}a^{8}-\frac{34\!\cdots\!17}{16\!\cdots\!72}a^{7}-\frac{62\!\cdots\!47}{16\!\cdots\!72}a^{6}+\frac{23\!\cdots\!11}{84\!\cdots\!36}a^{5}+\frac{56\!\cdots\!57}{42\!\cdots\!68}a^{4}+\frac{51\!\cdots\!07}{16\!\cdots\!72}a^{3}+\frac{10\!\cdots\!21}{16\!\cdots\!72}a^{2}+\frac{13\!\cdots\!86}{10\!\cdots\!67}a+\frac{17\!\cdots\!65}{10\!\cdots\!67}$, $\frac{15\!\cdots\!17}{16\!\cdots\!72}a^{13}-\frac{73\!\cdots\!33}{84\!\cdots\!36}a^{12}-\frac{16\!\cdots\!61}{16\!\cdots\!72}a^{11}-\frac{21\!\cdots\!89}{16\!\cdots\!72}a^{10}+\frac{83\!\cdots\!37}{16\!\cdots\!72}a^{9}+\frac{18\!\cdots\!10}{10\!\cdots\!67}a^{8}-\frac{84\!\cdots\!61}{84\!\cdots\!36}a^{7}-\frac{48\!\cdots\!63}{84\!\cdots\!36}a^{6}+\frac{19\!\cdots\!35}{21\!\cdots\!34}a^{5}+\frac{12\!\cdots\!91}{10\!\cdots\!67}a^{4}+\frac{26\!\cdots\!97}{84\!\cdots\!36}a^{3}+\frac{45\!\cdots\!33}{84\!\cdots\!36}a^{2}+\frac{58\!\cdots\!29}{42\!\cdots\!68}a+\frac{22\!\cdots\!58}{10\!\cdots\!67}$ (assuming GRH) | 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: | \( 8415941567.41 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 8415941567.41 \cdot 21}{2\cdot\sqrt{337337631352558562817384077600704}}\cr\approx \mathstrut & 1.86002462311 \end{aligned}\] (assuming GRH)
Galois group
$C_7^2:S_3$ (as 14T15):
A solvable group of order 294 |
The 20 conjugacy class representatives for $C_7^2:S_3$ |
Character table for $C_7^2:S_3$ |
Intermediate fields
\(\Q(\sqrt{-31}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 21 siblings: | data not computed |
Degree 42 siblings: | 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 | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.3.0.1}{3} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.14.0.1}{14} }$ | ${\href{/padicField/13.14.0.1}{14} }$ | ${\href{/padicField/17.14.0.1}{14} }$ | ${\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.14.0.1}{14} }$ | ${\href{/padicField/29.14.0.1}{14} }$ | R | ${\href{/padicField/37.14.0.1}{14} }$ | ${\href{/padicField/41.3.0.1}{3} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }$ | ${\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
\(7\) | 7.7.12.8 | $x^{7} + 21 x^{6} + 7$ | $7$ | $1$ | $12$ | $C_7:C_3$ | $[2]^{3}$ |
7.7.12.9 | $x^{7} + 35 x^{6} + 7$ | $7$ | $1$ | $12$ | $C_7:C_3$ | $[2]^{3}$ | |
\(31\) | 31.14.7.2 | $x^{14} + 217 x^{12} + 20181 x^{10} + 1042687 x^{8} + 56 x^{7} + 32322367 x^{6} - 36456 x^{5} + 601212171 x^{4} + 1883560 x^{3} + 6213359916 x^{2} - 11678016 x + 27510767884$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ |