Normalized defining polynomial
\( x^{14} - x^{13} - 7 x^{12} + 9 x^{11} + 25 x^{10} + 3 x^{9} - 117 x^{8} - 73 x^{7} + 406 x^{6} + \cdots - 116 \)
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: | \(4156382630830772224\) \(\medspace = 2^{12}\cdot 317^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(21.38\) | 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: | $2^{6/7}317^{1/2}\approx 32.251902756545775$ | ||
Ramified primes: | \(2\), \(317\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{4}$, $\frac{1}{18\!\cdots\!38}a^{13}+\frac{218102613586431}{18\!\cdots\!38}a^{12}+\frac{187073085814668}{945937402498519}a^{11}-\frac{191810521200383}{945937402498519}a^{10}+\frac{165760374023856}{945937402498519}a^{9}+\frac{339970133997857}{18\!\cdots\!38}a^{8}+\frac{455553928093765}{945937402498519}a^{7}-\frac{322970410798278}{945937402498519}a^{6}+\frac{835027717449179}{18\!\cdots\!38}a^{5}+\frac{35517148568566}{945937402498519}a^{4}+\frac{50546028537219}{945937402498519}a^{3}-\frac{12602477137948}{945937402498519}a^{2}-\frac{28314009263874}{945937402498519}a-\frac{172258960003668}{945937402498519}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
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: | $\frac{3574894745192}{945937402498519}a^{13}-\frac{2159374378552}{945937402498519}a^{12}-\frac{57677544141141}{18\!\cdots\!38}a^{11}+\frac{46720689034803}{18\!\cdots\!38}a^{10}+\frac{107157237967747}{945937402498519}a^{9}+\frac{76320734220127}{18\!\cdots\!38}a^{8}-\frac{811846558457025}{18\!\cdots\!38}a^{7}-\frac{10\!\cdots\!05}{18\!\cdots\!38}a^{6}+\frac{13\!\cdots\!65}{945937402498519}a^{5}+\frac{13\!\cdots\!05}{18\!\cdots\!38}a^{4}-\frac{16\!\cdots\!61}{945937402498519}a^{3}-\frac{67304835771584}{945937402498519}a^{2}+\frac{567097843674363}{945937402498519}a+\frac{306134875727733}{945937402498519}$, $\frac{957661712547}{945937402498519}a^{13}+\frac{5275995540949}{18\!\cdots\!38}a^{12}-\frac{15636673627705}{18\!\cdots\!38}a^{11}-\frac{9917507172162}{945937402498519}a^{10}+\frac{28581066965451}{945937402498519}a^{9}+\frac{70981188136198}{945937402498519}a^{8}-\frac{18271592148231}{18\!\cdots\!38}a^{7}-\frac{326606643241825}{945937402498519}a^{6}+\frac{64753987257224}{945937402498519}a^{5}+\frac{10\!\cdots\!69}{18\!\cdots\!38}a^{4}+\frac{32428368046210}{945937402498519}a^{3}-\frac{669774181820313}{945937402498519}a^{2}-\frac{228179050664449}{945937402498519}a+\frac{779210271022979}{945937402498519}$, $\frac{7225398409063}{18\!\cdots\!38}a^{13}+\frac{2519518034686}{945937402498519}a^{12}-\frac{50298910455335}{18\!\cdots\!38}a^{11}-\frac{18579203728937}{18\!\cdots\!38}a^{10}+\frac{107912299587647}{945937402498519}a^{9}+\frac{156748299948655}{945937402498519}a^{8}-\frac{508573532637545}{18\!\cdots\!38}a^{7}-\frac{14\!\cdots\!91}{18\!\cdots\!38}a^{6}+\frac{11\!\cdots\!85}{18\!\cdots\!38}a^{5}+\frac{15\!\cdots\!46}{945937402498519}a^{4}-\frac{783760350596467}{945937402498519}a^{3}-\frac{17\!\cdots\!85}{945937402498519}a^{2}-\frac{141712057130336}{945937402498519}a+\frac{708177475194611}{945937402498519}$, $\frac{3024223595452}{945937402498519}a^{13}+\frac{484039869569}{945937402498519}a^{12}-\frac{48776405255399}{18\!\cdots\!38}a^{11}+\frac{7740713633111}{18\!\cdots\!38}a^{10}+\frac{104530617332137}{945937402498519}a^{9}+\frac{155332385135995}{18\!\cdots\!38}a^{8}-\frac{617087231172421}{18\!\cdots\!38}a^{7}-\frac{10\!\cdots\!91}{18\!\cdots\!38}a^{6}+\frac{845324350434442}{945937402498519}a^{5}+\frac{22\!\cdots\!83}{18\!\cdots\!38}a^{4}-\frac{12\!\cdots\!13}{945937402498519}a^{3}-\frac{942995045594481}{945937402498519}a^{2}+\frac{269079349758297}{945937402498519}a+\frac{12\!\cdots\!11}{945937402498519}$, $\frac{5482295843469}{945937402498519}a^{13}-\frac{16256498450645}{18\!\cdots\!38}a^{12}-\frac{69940693757343}{18\!\cdots\!38}a^{11}+\frac{134214964332477}{18\!\cdots\!38}a^{10}+\frac{95616187687498}{945937402498519}a^{9}-\frac{20233453212933}{18\!\cdots\!38}a^{8}-\frac{12\!\cdots\!97}{18\!\cdots\!38}a^{7}-\frac{318242107628387}{18\!\cdots\!38}a^{6}+\frac{21\!\cdots\!15}{945937402498519}a^{5}-\frac{10\!\cdots\!88}{945937402498519}a^{4}-\frac{10\!\cdots\!67}{945937402498519}a^{3}-\frac{321167931422674}{945937402498519}a^{2}+\frac{17\!\cdots\!05}{945937402498519}a-\frac{10\!\cdots\!83}{945937402498519}$, $\frac{3575615051758}{945937402498519}a^{13}+\frac{3273072063224}{945937402498519}a^{12}-\frac{37846154854321}{18\!\cdots\!38}a^{11}-\frac{23605969103869}{18\!\cdots\!38}a^{10}+\frac{57980817956709}{945937402498519}a^{9}+\frac{325979405968723}{18\!\cdots\!38}a^{8}-\frac{108386685300629}{18\!\cdots\!38}a^{7}-\frac{975237923545985}{18\!\cdots\!38}a^{6}+\frac{37034342417405}{945937402498519}a^{5}+\frac{334984936615725}{18\!\cdots\!38}a^{4}-\frac{28558313018960}{945937402498519}a^{3}+\frac{141637557691127}{945937402498519}a^{2}+\frac{10\!\cdots\!21}{945937402498519}a+\frac{201515299181701}{945937402498519}$, $\frac{20350950245099}{18\!\cdots\!38}a^{13}-\frac{51391907308981}{18\!\cdots\!38}a^{12}-\frac{91076975812315}{18\!\cdots\!38}a^{11}+\frac{333569538713863}{18\!\cdots\!38}a^{10}+\frac{50739199427282}{945937402498519}a^{9}-\frac{130796490530113}{945937402498519}a^{8}-\frac{22\!\cdots\!83}{18\!\cdots\!38}a^{7}+\frac{12\!\cdots\!43}{18\!\cdots\!38}a^{6}+\frac{74\!\cdots\!95}{18\!\cdots\!38}a^{5}-\frac{10\!\cdots\!09}{18\!\cdots\!38}a^{4}+\frac{551239192071994}{945937402498519}a^{3}-\frac{253136302070753}{945937402498519}a^{2}+\frac{24\!\cdots\!86}{945937402498519}a-\frac{870717228943687}{945937402498519}$ | 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: | \( 5502.73317638 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 5502.73317638 \cdot 3}{2\cdot\sqrt{4156382630830772224}}\cr\approx \mathstrut & 0.996440003506 \end{aligned}\]
Galois group
$\GL(3,2)$ (as 14T10):
A non-solvable group of order 168 |
The 6 conjugacy class representatives for $\PSL(2,7)$ |
Character table for $\PSL(2,7)$ |
Intermediate fields
7.3.6431296.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 7 siblings: | 7.3.6431296.2, 7.3.6431296.1 |
Degree 8 sibling: | 8.0.646274503744.1 |
Degree 21 sibling: | 21.5.26730926988131422081122304.1 |
Degree 24 sibling: | data not computed |
Degree 28 sibling: | data not computed |
Degree 42 siblings: | data not computed |
Arithmetically equvalently sibling: | 14.2.4156382630830772224.2 |
Minimal sibling: | 7.3.6431296.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.7.0.1}{7} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.7.0.1}{7} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }^{2}$ | ${\href{/padicField/37.7.0.1}{7} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
\(317\) | $\Q_{317}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{317}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ |