Normalized defining polynomial
\( x^{14} - 4 x^{13} - 4 x^{12} + 32 x^{11} - 44 x^{10} + 76 x^{9} + 222 x^{8} - 1472 x^{7} + 483 x^{6} + \cdots - 1573 \)
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)
| |
Discriminant: | \(7490222540601799313\) \(\medspace = 7^{7}\cdot 71^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(22.29\) | 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: | $7^{1/2}71^{1/2}\approx 22.293496809607955$ | ||
Ramified primes: | \(7\), \(71\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{497}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{142}a^{12}+\frac{6}{71}a^{11}-\frac{9}{71}a^{10}-\frac{15}{71}a^{9}-\frac{11}{142}a^{8}-\frac{13}{71}a^{7}-\frac{29}{71}a^{6}+\frac{45}{142}a^{5}+\frac{8}{71}a^{4}-\frac{5}{71}a^{3}-\frac{20}{71}a^{2}+\frac{1}{71}a+\frac{16}{71}$, $\frac{1}{10\!\cdots\!86}a^{13}+\frac{21\!\cdots\!75}{10\!\cdots\!86}a^{12}+\frac{12\!\cdots\!05}{10\!\cdots\!86}a^{11}-\frac{79\!\cdots\!11}{10\!\cdots\!86}a^{10}-\frac{55\!\cdots\!72}{46\!\cdots\!63}a^{9}+\frac{11\!\cdots\!24}{51\!\cdots\!93}a^{8}+\frac{17\!\cdots\!21}{10\!\cdots\!86}a^{7}-\frac{77\!\cdots\!60}{51\!\cdots\!93}a^{6}-\frac{18\!\cdots\!30}{51\!\cdots\!93}a^{5}-\frac{97\!\cdots\!21}{51\!\cdots\!93}a^{4}-\frac{30\!\cdots\!05}{10\!\cdots\!86}a^{3}-\frac{13\!\cdots\!45}{10\!\cdots\!86}a^{2}+\frac{17\!\cdots\!53}{10\!\cdots\!86}a+\frac{21\!\cdots\!30}{46\!\cdots\!63}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{78\!\cdots\!07}{10\!\cdots\!86}a^{13}-\frac{11\!\cdots\!23}{51\!\cdots\!93}a^{12}-\frac{22\!\cdots\!51}{51\!\cdots\!93}a^{11}+\frac{19\!\cdots\!89}{10\!\cdots\!86}a^{10}-\frac{21\!\cdots\!29}{93\!\cdots\!26}a^{9}+\frac{51\!\cdots\!83}{10\!\cdots\!86}a^{8}+\frac{20\!\cdots\!83}{10\!\cdots\!86}a^{7}-\frac{45\!\cdots\!56}{51\!\cdots\!93}a^{6}-\frac{85\!\cdots\!46}{51\!\cdots\!93}a^{5}+\frac{25\!\cdots\!45}{10\!\cdots\!86}a^{4}-\frac{64\!\cdots\!96}{51\!\cdots\!93}a^{3}-\frac{97\!\cdots\!40}{51\!\cdots\!93}a^{2}-\frac{15\!\cdots\!73}{10\!\cdots\!86}a+\frac{80\!\cdots\!35}{93\!\cdots\!26}$, $\frac{16\!\cdots\!33}{10\!\cdots\!86}a^{13}-\frac{50\!\cdots\!15}{10\!\cdots\!86}a^{12}+\frac{12\!\cdots\!71}{10\!\cdots\!86}a^{11}+\frac{95\!\cdots\!96}{51\!\cdots\!93}a^{10}-\frac{42\!\cdots\!93}{46\!\cdots\!63}a^{9}+\frac{10\!\cdots\!01}{51\!\cdots\!93}a^{8}-\frac{34\!\cdots\!57}{10\!\cdots\!86}a^{7}-\frac{11\!\cdots\!85}{10\!\cdots\!86}a^{6}+\frac{22\!\cdots\!75}{51\!\cdots\!93}a^{5}-\frac{68\!\cdots\!11}{51\!\cdots\!93}a^{4}-\frac{45\!\cdots\!97}{51\!\cdots\!93}a^{3}+\frac{39\!\cdots\!69}{10\!\cdots\!86}a^{2}+\frac{36\!\cdots\!09}{10\!\cdots\!86}a+\frac{17\!\cdots\!95}{46\!\cdots\!63}$, $\frac{739927366893841}{10\!\cdots\!86}a^{13}-\frac{11\!\cdots\!33}{10\!\cdots\!86}a^{12}-\frac{30\!\cdots\!11}{10\!\cdots\!86}a^{11}+\frac{99\!\cdots\!39}{51\!\cdots\!93}a^{10}+\frac{37\!\cdots\!19}{93\!\cdots\!26}a^{9}+\frac{21\!\cdots\!52}{51\!\cdots\!93}a^{8}-\frac{93\!\cdots\!03}{10\!\cdots\!86}a^{7}-\frac{23\!\cdots\!31}{10\!\cdots\!86}a^{6}-\frac{16\!\cdots\!33}{10\!\cdots\!86}a^{5}+\frac{88\!\cdots\!91}{51\!\cdots\!93}a^{4}+\frac{52\!\cdots\!33}{51\!\cdots\!93}a^{3}-\frac{16\!\cdots\!88}{51\!\cdots\!93}a^{2}-\frac{16\!\cdots\!33}{10\!\cdots\!86}a-\frac{47\!\cdots\!27}{46\!\cdots\!63}$, $\frac{49\!\cdots\!39}{10\!\cdots\!86}a^{13}-\frac{22\!\cdots\!07}{10\!\cdots\!86}a^{12}-\frac{43\!\cdots\!59}{10\!\cdots\!86}a^{11}+\frac{89\!\cdots\!71}{51\!\cdots\!93}a^{10}-\frac{24\!\cdots\!90}{46\!\cdots\!63}a^{9}+\frac{24\!\cdots\!83}{51\!\cdots\!93}a^{8}+\frac{14\!\cdots\!45}{10\!\cdots\!86}a^{7}-\frac{89\!\cdots\!71}{10\!\cdots\!86}a^{6}-\frac{37\!\cdots\!47}{51\!\cdots\!93}a^{5}+\frac{11\!\cdots\!34}{51\!\cdots\!93}a^{4}+\frac{89\!\cdots\!00}{51\!\cdots\!93}a^{3}-\frac{12\!\cdots\!89}{14\!\cdots\!66}a^{2}-\frac{43\!\cdots\!47}{10\!\cdots\!86}a+\frac{40\!\cdots\!16}{46\!\cdots\!63}$, $\frac{34\!\cdots\!70}{51\!\cdots\!93}a^{13}+\frac{35\!\cdots\!39}{51\!\cdots\!93}a^{12}-\frac{29\!\cdots\!97}{51\!\cdots\!93}a^{11}-\frac{31\!\cdots\!09}{51\!\cdots\!93}a^{10}-\frac{49\!\cdots\!90}{46\!\cdots\!63}a^{9}+\frac{21\!\cdots\!72}{51\!\cdots\!93}a^{8}+\frac{28\!\cdots\!19}{10\!\cdots\!86}a^{7}+\frac{58\!\cdots\!27}{51\!\cdots\!93}a^{6}-\frac{50\!\cdots\!32}{51\!\cdots\!93}a^{5}-\frac{99\!\cdots\!88}{51\!\cdots\!93}a^{4}+\frac{17\!\cdots\!15}{10\!\cdots\!86}a^{3}+\frac{19\!\cdots\!37}{51\!\cdots\!93}a^{2}+\frac{21\!\cdots\!31}{51\!\cdots\!93}a+\frac{14\!\cdots\!89}{93\!\cdots\!26}$, $\frac{16\!\cdots\!01}{10\!\cdots\!86}a^{13}-\frac{92\!\cdots\!45}{10\!\cdots\!86}a^{12}+\frac{80\!\cdots\!33}{10\!\cdots\!86}a^{11}+\frac{40\!\cdots\!59}{10\!\cdots\!86}a^{10}-\frac{60\!\cdots\!58}{46\!\cdots\!63}a^{9}+\frac{16\!\cdots\!57}{51\!\cdots\!93}a^{8}-\frac{17\!\cdots\!83}{10\!\cdots\!86}a^{7}-\frac{10\!\cdots\!42}{51\!\cdots\!93}a^{6}+\frac{20\!\cdots\!43}{51\!\cdots\!93}a^{5}+\frac{55\!\cdots\!97}{51\!\cdots\!93}a^{4}-\frac{42\!\cdots\!63}{10\!\cdots\!86}a^{3}-\frac{23\!\cdots\!59}{10\!\cdots\!86}a^{2}+\frac{30\!\cdots\!13}{10\!\cdots\!86}a-\frac{49\!\cdots\!29}{46\!\cdots\!63}$, $\frac{28\!\cdots\!76}{51\!\cdots\!93}a^{13}-\frac{65\!\cdots\!13}{10\!\cdots\!86}a^{12}-\frac{47\!\cdots\!01}{10\!\cdots\!86}a^{11}+\frac{25\!\cdots\!51}{51\!\cdots\!93}a^{10}-\frac{36\!\cdots\!75}{93\!\cdots\!26}a^{9}+\frac{22\!\cdots\!37}{10\!\cdots\!86}a^{8}+\frac{94\!\cdots\!92}{51\!\cdots\!93}a^{7}-\frac{16\!\cdots\!77}{51\!\cdots\!93}a^{6}-\frac{36\!\cdots\!19}{51\!\cdots\!93}a^{5}+\frac{85\!\cdots\!15}{10\!\cdots\!86}a^{4}+\frac{11\!\cdots\!85}{10\!\cdots\!86}a^{3}-\frac{32\!\cdots\!07}{10\!\cdots\!86}a^{2}-\frac{25\!\cdots\!51}{51\!\cdots\!93}a+\frac{33\!\cdots\!35}{93\!\cdots\!26}$ | 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: | \( 6632.36500949 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 6632.36500949 \cdot 1}{2\cdot\sqrt{7490222540601799313}}\cr\approx \mathstrut & 0.298215535804 \end{aligned}\]
Galois group
A solvable group of order 28 |
The 10 conjugacy class representatives for $D_{14}$ |
Character table for $D_{14}$ |
Intermediate fields
\(\Q(\sqrt{497}) \), 7.1.357911.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 28 |
Degree 14 sibling: | 14.0.105496092121152103.1 |
Minimal sibling: | 14.0.105496092121152103.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.14.0.1}{14} }$ | R | ${\href{/padicField/11.2.0.1}{2} }^{7}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.14.0.1}{14} }$ | ${\href{/padicField/23.2.0.1}{2} }^{7}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.7.0.1}{7} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{7}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(71\) | 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.497.2t1.a.a | $1$ | $ 7 \cdot 71 $ | \(\Q(\sqrt{497}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.71.2t1.a.a | $1$ | $ 71 $ | \(\Q(\sqrt{-71}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.3479.14t3.a.b | $2$ | $ 7^{2} \cdot 71 $ | 14.2.7490222540601799313.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
* | 2.71.7t2.a.c | $2$ | $ 71 $ | 7.1.357911.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.71.7t2.a.b | $2$ | $ 71 $ | 7.1.357911.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.71.7t2.a.a | $2$ | $ 71 $ | 7.1.357911.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.3479.14t3.a.c | $2$ | $ 7^{2} \cdot 71 $ | 14.2.7490222540601799313.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
* | 2.3479.14t3.a.a | $2$ | $ 7^{2} \cdot 71 $ | 14.2.7490222540601799313.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |