Normalized defining polynomial
\( x^{14} - 2 x^{13} - 23 x^{12} + 92 x^{11} - 124 x^{10} + 200 x^{9} - 348 x^{8} + 1188 x^{7} + \cdots - 10607 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(127832656740959562315595776\) \(\medspace = 2^{27}\cdot 3^{12}\cdot 13^{11}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(73.24\) | 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: | $2^{19/8}3^{6/7}13^{5/6}\approx 112.76996407954955$ | ||
Ramified primes: | \(2\), \(3\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{26}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{3}$, $\frac{1}{16\!\cdots\!36}a^{13}+\frac{64\!\cdots\!79}{16\!\cdots\!36}a^{12}+\frac{25\!\cdots\!01}{40\!\cdots\!09}a^{11}-\frac{13\!\cdots\!39}{40\!\cdots\!09}a^{10}+\frac{10\!\cdots\!50}{13\!\cdots\!03}a^{9}-\frac{53\!\cdots\!05}{40\!\cdots\!09}a^{8}-\frac{11\!\cdots\!96}{13\!\cdots\!03}a^{7}-\frac{75\!\cdots\!29}{40\!\cdots\!09}a^{6}+\frac{35\!\cdots\!21}{16\!\cdots\!36}a^{5}+\frac{44\!\cdots\!31}{16\!\cdots\!36}a^{4}+\frac{55\!\cdots\!35}{40\!\cdots\!09}a^{3}+\frac{69\!\cdots\!26}{40\!\cdots\!09}a^{2}-\frac{14\!\cdots\!91}{16\!\cdots\!36}a-\frac{71\!\cdots\!33}{16\!\cdots\!36}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{4}$, which has order $8$ (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: | $\frac{58\!\cdots\!70}{40\!\cdots\!09}a^{13}+\frac{50\!\cdots\!73}{13\!\cdots\!03}a^{12}+\frac{12\!\cdots\!16}{40\!\cdots\!09}a^{11}-\frac{63\!\cdots\!20}{40\!\cdots\!09}a^{10}+\frac{11\!\cdots\!42}{40\!\cdots\!09}a^{9}-\frac{44\!\cdots\!46}{13\!\cdots\!03}a^{8}+\frac{54\!\cdots\!32}{13\!\cdots\!03}a^{7}-\frac{25\!\cdots\!58}{13\!\cdots\!03}a^{6}+\frac{10\!\cdots\!24}{13\!\cdots\!03}a^{5}-\frac{58\!\cdots\!14}{40\!\cdots\!09}a^{4}+\frac{14\!\cdots\!46}{13\!\cdots\!03}a^{3}-\frac{11\!\cdots\!66}{40\!\cdots\!09}a^{2}+\frac{38\!\cdots\!14}{40\!\cdots\!09}a+\frac{38\!\cdots\!08}{40\!\cdots\!09}$, $\frac{41\!\cdots\!83}{40\!\cdots\!09}a^{13}+\frac{13\!\cdots\!99}{13\!\cdots\!03}a^{12}+\frac{10\!\cdots\!57}{40\!\cdots\!09}a^{11}-\frac{26\!\cdots\!60}{40\!\cdots\!09}a^{10}+\frac{26\!\cdots\!98}{40\!\cdots\!09}a^{9}-\frac{19\!\cdots\!38}{13\!\cdots\!03}a^{8}+\frac{37\!\cdots\!17}{13\!\cdots\!03}a^{7}-\frac{87\!\cdots\!26}{13\!\cdots\!03}a^{6}+\frac{59\!\cdots\!88}{13\!\cdots\!03}a^{5}-\frac{95\!\cdots\!12}{40\!\cdots\!09}a^{4}-\frac{11\!\cdots\!52}{13\!\cdots\!03}a^{3}-\frac{34\!\cdots\!31}{40\!\cdots\!09}a^{2}-\frac{19\!\cdots\!37}{40\!\cdots\!09}a-\frac{25\!\cdots\!73}{40\!\cdots\!09}$, $\frac{68\!\cdots\!49}{40\!\cdots\!09}a^{13}+\frac{88\!\cdots\!96}{13\!\cdots\!03}a^{12}+\frac{14\!\cdots\!49}{40\!\cdots\!09}a^{11}-\frac{31\!\cdots\!88}{13\!\cdots\!03}a^{10}+\frac{18\!\cdots\!48}{40\!\cdots\!09}a^{9}-\frac{18\!\cdots\!26}{40\!\cdots\!09}a^{8}+\frac{27\!\cdots\!59}{40\!\cdots\!09}a^{7}-\frac{10\!\cdots\!76}{40\!\cdots\!09}a^{6}+\frac{46\!\cdots\!12}{40\!\cdots\!09}a^{5}-\frac{99\!\cdots\!62}{40\!\cdots\!09}a^{4}+\frac{83\!\cdots\!62}{40\!\cdots\!09}a^{3}+\frac{63\!\cdots\!83}{40\!\cdots\!09}a^{2}+\frac{22\!\cdots\!73}{40\!\cdots\!09}a-\frac{11\!\cdots\!66}{40\!\cdots\!09}$, $\frac{77\!\cdots\!13}{40\!\cdots\!09}a^{13}+\frac{88\!\cdots\!96}{40\!\cdots\!09}a^{12}+\frac{60\!\cdots\!04}{13\!\cdots\!03}a^{11}-\frac{18\!\cdots\!11}{13\!\cdots\!03}a^{10}+\frac{53\!\cdots\!72}{40\!\cdots\!09}a^{9}-\frac{14\!\cdots\!25}{40\!\cdots\!09}a^{8}+\frac{24\!\cdots\!33}{40\!\cdots\!09}a^{7}-\frac{65\!\cdots\!42}{40\!\cdots\!09}a^{6}+\frac{28\!\cdots\!39}{40\!\cdots\!09}a^{5}-\frac{15\!\cdots\!58}{13\!\cdots\!03}a^{4}+\frac{84\!\cdots\!31}{13\!\cdots\!03}a^{3}-\frac{24\!\cdots\!81}{40\!\cdots\!09}a^{2}-\frac{65\!\cdots\!28}{40\!\cdots\!09}a+\frac{15\!\cdots\!02}{40\!\cdots\!09}$, $\frac{13\!\cdots\!80}{40\!\cdots\!09}a^{13}+\frac{76\!\cdots\!42}{40\!\cdots\!09}a^{12}+\frac{29\!\cdots\!24}{40\!\cdots\!09}a^{11}-\frac{26\!\cdots\!28}{13\!\cdots\!03}a^{10}+\frac{98\!\cdots\!88}{40\!\cdots\!09}a^{9}-\frac{21\!\cdots\!50}{40\!\cdots\!09}a^{8}+\frac{17\!\cdots\!56}{40\!\cdots\!09}a^{7}-\frac{14\!\cdots\!88}{40\!\cdots\!09}a^{6}+\frac{41\!\cdots\!72}{40\!\cdots\!09}a^{5}-\frac{20\!\cdots\!27}{13\!\cdots\!03}a^{4}+\frac{23\!\cdots\!36}{40\!\cdots\!09}a^{3}-\frac{56\!\cdots\!16}{40\!\cdots\!09}a^{2}-\frac{22\!\cdots\!32}{40\!\cdots\!09}a-\frac{22\!\cdots\!16}{13\!\cdots\!03}$, $\frac{17\!\cdots\!79}{40\!\cdots\!09}a^{13}+\frac{12\!\cdots\!50}{40\!\cdots\!09}a^{12}+\frac{29\!\cdots\!81}{40\!\cdots\!09}a^{11}-\frac{38\!\cdots\!38}{40\!\cdots\!09}a^{10}+\frac{91\!\cdots\!56}{40\!\cdots\!09}a^{9}-\frac{54\!\cdots\!85}{40\!\cdots\!09}a^{8}+\frac{60\!\cdots\!17}{40\!\cdots\!09}a^{7}-\frac{54\!\cdots\!48}{40\!\cdots\!09}a^{6}+\frac{21\!\cdots\!54}{40\!\cdots\!09}a^{5}-\frac{15\!\cdots\!86}{13\!\cdots\!03}a^{4}+\frac{13\!\cdots\!64}{13\!\cdots\!03}a^{3}+\frac{18\!\cdots\!59}{13\!\cdots\!03}a^{2}+\frac{40\!\cdots\!91}{13\!\cdots\!03}a-\frac{23\!\cdots\!07}{13\!\cdots\!03}$, $\frac{55\!\cdots\!67}{16\!\cdots\!36}a^{13}+\frac{39\!\cdots\!91}{54\!\cdots\!12}a^{12}-\frac{31\!\cdots\!65}{40\!\cdots\!09}a^{11}+\frac{58\!\cdots\!43}{40\!\cdots\!09}a^{10}-\frac{14\!\cdots\!59}{13\!\cdots\!03}a^{9}+\frac{18\!\cdots\!19}{40\!\cdots\!09}a^{8}-\frac{25\!\cdots\!16}{13\!\cdots\!03}a^{7}+\frac{14\!\cdots\!09}{40\!\cdots\!09}a^{6}-\frac{13\!\cdots\!37}{16\!\cdots\!36}a^{5}+\frac{13\!\cdots\!81}{16\!\cdots\!36}a^{4}+\frac{71\!\cdots\!94}{40\!\cdots\!09}a^{3}+\frac{25\!\cdots\!73}{13\!\cdots\!03}a^{2}+\frac{16\!\cdots\!51}{16\!\cdots\!36}a+\frac{26\!\cdots\!37}{16\!\cdots\!36}$ (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: | \( 6671455.421958637 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 6671455.421958637 \cdot 8}{2\cdot\sqrt{127832656740959562315595776}}\cr\approx \mathstrut & 0.580896652488332 \end{aligned}\] (assuming GRH)
Galois group
$C_2^6:F_7$ (as 14T40):
A solvable group of order 2688 |
The 20 conjugacy class representatives for $C_2^6:F_7$ |
Character table for $C_2^6:F_7$ |
Intermediate fields
7.7.138584369664.1 |
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 16 sibling: | data not computed |
Degree 28 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 | R | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.12.24.107 | $x^{12} + 12 x^{11} + 48 x^{10} + 64 x^{9} + 86 x^{8} + 488 x^{7} + 1000 x^{6} + 320 x^{5} + 1044 x^{4} + 672 x^{3} + 576 x^{2} + 224 x + 88$ | $4$ | $3$ | $24$ | $C_2^2 \times A_4$ | $[2, 2, 2, 3]^{3}$ | |
\(3\) | 3.14.12.1 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(13\) | 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.6.5.5 | $x^{6} + 65$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
13.6.5.5 | $x^{6} + 65$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |