Normalized defining polynomial
\( x^{14} - 3 x^{13} + 27 x^{12} - 147 x^{11} + 675 x^{10} - 6441 x^{9} - 13731 x^{8} + 86793 x^{7} + \cdots - 61742484 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(14655083487531544887678406656\) \(\medspace = 2^{12}\cdot 3^{12}\cdot 7^{11}\cdot 23^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(102.77\) | 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: | $2^{6/7}3^{6/7}7^{5/6}23^{1/2}\approx 112.74546204229466$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{161}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{3}a^{8}$, $\frac{1}{3}a^{9}$, $\frac{1}{3}a^{10}$, $\frac{1}{6}a^{11}-\frac{1}{6}a^{10}-\frac{1}{6}a^{9}-\frac{1}{6}a^{8}-\frac{1}{6}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{78}a^{12}+\frac{1}{78}a^{11}-\frac{5}{78}a^{10}+\frac{5}{78}a^{9}-\frac{3}{26}a^{8}-\frac{1}{78}a^{7}-\frac{7}{26}a^{6}+\frac{3}{26}a^{5}-\frac{1}{13}a^{4}-\frac{6}{13}a^{3}+\frac{6}{13}a^{2}-\frac{3}{13}a-\frac{3}{13}$, $\frac{1}{73\!\cdots\!74}a^{13}+\frac{38\!\cdots\!98}{12\!\cdots\!29}a^{12}-\frac{41\!\cdots\!46}{40\!\cdots\!43}a^{11}+\frac{42\!\cdots\!99}{40\!\cdots\!43}a^{10}-\frac{16\!\cdots\!33}{12\!\cdots\!29}a^{9}+\frac{41\!\cdots\!60}{40\!\cdots\!43}a^{8}-\frac{24\!\cdots\!59}{40\!\cdots\!43}a^{7}+\frac{58\!\cdots\!13}{12\!\cdots\!29}a^{6}-\frac{56\!\cdots\!71}{81\!\cdots\!86}a^{5}-\frac{91\!\cdots\!05}{40\!\cdots\!43}a^{4}+\frac{15\!\cdots\!79}{40\!\cdots\!43}a^{3}+\frac{20\!\cdots\!33}{40\!\cdots\!43}a^{2}+\frac{18\!\cdots\!42}{40\!\cdots\!43}a-\frac{16\!\cdots\!67}{40\!\cdots\!43}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{7}$, which has order $7$ (assuming GRH)
Unit group
Rank: | $7$ | 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: | $\frac{13\!\cdots\!37}{12\!\cdots\!29}a^{13}-\frac{30\!\cdots\!27}{24\!\cdots\!58}a^{12}+\frac{52\!\cdots\!21}{18\!\cdots\!66}a^{11}-\frac{96\!\cdots\!65}{81\!\cdots\!86}a^{10}+\frac{13\!\cdots\!87}{24\!\cdots\!58}a^{9}-\frac{52\!\cdots\!99}{81\!\cdots\!86}a^{8}-\frac{16\!\cdots\!95}{63\!\cdots\!22}a^{7}+\frac{32\!\cdots\!43}{81\!\cdots\!86}a^{6}+\frac{62\!\cdots\!03}{63\!\cdots\!22}a^{5}+\frac{17\!\cdots\!40}{40\!\cdots\!43}a^{4}+\frac{15\!\cdots\!59}{40\!\cdots\!43}a^{3}-\frac{19\!\cdots\!61}{40\!\cdots\!43}a^{2}+\frac{12\!\cdots\!03}{40\!\cdots\!43}a+\frac{12\!\cdots\!13}{40\!\cdots\!43}$, $\frac{13\!\cdots\!03}{24\!\cdots\!58}a^{13}-\frac{60\!\cdots\!67}{40\!\cdots\!43}a^{12}+\frac{17\!\cdots\!93}{12\!\cdots\!29}a^{11}-\frac{31\!\cdots\!96}{40\!\cdots\!43}a^{10}+\frac{45\!\cdots\!25}{12\!\cdots\!29}a^{9}-\frac{31\!\cdots\!45}{94\!\cdots\!33}a^{8}-\frac{27\!\cdots\!33}{40\!\cdots\!43}a^{7}+\frac{17\!\cdots\!20}{40\!\cdots\!43}a^{6}+\frac{23\!\cdots\!99}{81\!\cdots\!86}a^{5}+\frac{79\!\cdots\!48}{40\!\cdots\!43}a^{4}-\frac{15\!\cdots\!53}{40\!\cdots\!43}a^{3}-\frac{10\!\cdots\!41}{40\!\cdots\!43}a^{2}+\frac{18\!\cdots\!37}{40\!\cdots\!43}a+\frac{12\!\cdots\!57}{40\!\cdots\!43}$, $\frac{53\!\cdots\!21}{24\!\cdots\!58}a^{13}-\frac{11\!\cdots\!91}{12\!\cdots\!29}a^{12}+\frac{13\!\cdots\!79}{24\!\cdots\!58}a^{11}-\frac{15\!\cdots\!33}{81\!\cdots\!86}a^{10}+\frac{23\!\cdots\!41}{24\!\cdots\!58}a^{9}-\frac{28\!\cdots\!27}{24\!\cdots\!58}a^{8}-\frac{47\!\cdots\!73}{81\!\cdots\!86}a^{7}+\frac{37\!\cdots\!93}{81\!\cdots\!86}a^{6}+\frac{98\!\cdots\!98}{40\!\cdots\!43}a^{5}+\frac{68\!\cdots\!65}{81\!\cdots\!86}a^{4}+\frac{26\!\cdots\!58}{40\!\cdots\!43}a^{3}-\frac{36\!\cdots\!96}{40\!\cdots\!43}a^{2}+\frac{62\!\cdots\!07}{40\!\cdots\!43}a+\frac{11\!\cdots\!41}{40\!\cdots\!43}$, $\frac{16\!\cdots\!05}{73\!\cdots\!74}a^{13}-\frac{57\!\cdots\!81}{81\!\cdots\!86}a^{12}+\frac{76\!\cdots\!94}{12\!\cdots\!29}a^{11}-\frac{21\!\cdots\!47}{12\!\cdots\!29}a^{10}+\frac{14\!\cdots\!45}{12\!\cdots\!29}a^{9}-\frac{49\!\cdots\!81}{40\!\cdots\!43}a^{8}-\frac{75\!\cdots\!72}{12\!\cdots\!29}a^{7}+\frac{14\!\cdots\!25}{12\!\cdots\!29}a^{6}-\frac{23\!\cdots\!13}{81\!\cdots\!86}a^{5}+\frac{69\!\cdots\!13}{81\!\cdots\!86}a^{4}+\frac{27\!\cdots\!38}{40\!\cdots\!43}a^{3}-\frac{28\!\cdots\!98}{40\!\cdots\!43}a^{2}+\frac{27\!\cdots\!54}{40\!\cdots\!43}a+\frac{19\!\cdots\!53}{40\!\cdots\!43}$, $\frac{41\!\cdots\!11}{12\!\cdots\!29}a^{13}+\frac{12\!\cdots\!71}{24\!\cdots\!58}a^{12}+\frac{12\!\cdots\!64}{12\!\cdots\!29}a^{11}-\frac{62\!\cdots\!53}{40\!\cdots\!43}a^{10}+\frac{70\!\cdots\!49}{40\!\cdots\!43}a^{9}-\frac{16\!\cdots\!88}{12\!\cdots\!29}a^{8}-\frac{15\!\cdots\!19}{12\!\cdots\!29}a^{7}-\frac{80\!\cdots\!72}{40\!\cdots\!43}a^{6}-\frac{41\!\cdots\!36}{40\!\cdots\!43}a^{5}+\frac{72\!\cdots\!19}{81\!\cdots\!86}a^{4}+\frac{73\!\cdots\!12}{40\!\cdots\!43}a^{3}-\frac{37\!\cdots\!88}{40\!\cdots\!43}a^{2}+\frac{28\!\cdots\!98}{40\!\cdots\!43}a+\frac{19\!\cdots\!95}{40\!\cdots\!43}$, $\frac{36\!\cdots\!16}{12\!\cdots\!29}a^{13}+\frac{20\!\cdots\!69}{81\!\cdots\!86}a^{12}+\frac{22\!\cdots\!49}{81\!\cdots\!86}a^{11}-\frac{71\!\cdots\!71}{81\!\cdots\!86}a^{10}-\frac{15\!\cdots\!71}{24\!\cdots\!58}a^{9}+\frac{28\!\cdots\!11}{24\!\cdots\!58}a^{8}+\frac{21\!\cdots\!93}{24\!\cdots\!58}a^{7}+\frac{40\!\cdots\!65}{81\!\cdots\!86}a^{6}+\frac{30\!\cdots\!75}{81\!\cdots\!86}a^{5}-\frac{35\!\cdots\!97}{40\!\cdots\!43}a^{4}-\frac{92\!\cdots\!05}{40\!\cdots\!43}a^{3}+\frac{21\!\cdots\!65}{40\!\cdots\!43}a^{2}-\frac{24\!\cdots\!33}{40\!\cdots\!43}a-\frac{36\!\cdots\!27}{40\!\cdots\!43}$, $\frac{10\!\cdots\!03}{73\!\cdots\!74}a^{13}-\frac{46\!\cdots\!54}{40\!\cdots\!43}a^{12}+\frac{10\!\cdots\!50}{12\!\cdots\!29}a^{11}-\frac{69\!\cdots\!61}{12\!\cdots\!29}a^{10}+\frac{40\!\cdots\!76}{12\!\cdots\!29}a^{9}-\frac{91\!\cdots\!29}{40\!\cdots\!43}a^{8}+\frac{84\!\cdots\!94}{12\!\cdots\!29}a^{7}-\frac{11\!\cdots\!51}{12\!\cdots\!29}a^{6}+\frac{95\!\cdots\!71}{81\!\cdots\!86}a^{5}+\frac{22\!\cdots\!76}{40\!\cdots\!43}a^{4}-\frac{15\!\cdots\!77}{40\!\cdots\!43}a^{3}+\frac{35\!\cdots\!95}{40\!\cdots\!43}a^{2}-\frac{35\!\cdots\!82}{40\!\cdots\!43}a+\frac{80\!\cdots\!65}{31\!\cdots\!11}$ (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: | \( 198701591.36121362 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 198701591.36121362 \cdot 7}{2\cdot\sqrt{14655083487531544887678406656}}\cr\approx \mathstrut & 1.41388615782381 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times F_7$ (as 14T7):
A solvable group of order 84 |
The 14 conjugacy class representatives for $F_7 \times C_2$ |
Character table for $F_7 \times C_2$ |
Intermediate fields
\(\Q(\sqrt{161}) \), 7.1.784147392.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 14 sibling: | data not computed |
Degree 28 sibling: | data not computed |
Degree 42 siblings: | data not computed |
Minimal sibling: | 14.0.2093583355361649269668343808.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
\(3\) | 3.14.12.1 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(7\) | 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ | |
\(23\) | 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
23.6.3.2 | $x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
23.6.3.2 | $x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |