Normalized defining polynomial
\( x^{14} - x^{13} - 15 x^{12} + 17 x^{11} + 59 x^{10} - 23 x^{9} - 107 x^{8} - 435 x^{7} + 594 x^{6} + 388 x^{5} + 1158 x^{4} - 408 x^{3} + 250 x^{2} + 6 x + 2 \)
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)
| |
Discriminant: | \(-1977326743000000000000\) \(\medspace = -\,2^{12}\cdot 5^{12}\cdot 7^{11}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(33.20\) | 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: | $2^{6/7}5^{6/7}7^{5/6}\approx 36.42430080048362$ | ||
Ramified primes: | \(2\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{11}a^{11}+\frac{2}{11}a^{10}-\frac{4}{11}a^{9}-\frac{1}{11}a^{8}+\frac{4}{11}a^{7}+\frac{4}{11}a^{6}-\frac{4}{11}a^{5}+\frac{4}{11}a^{4}+\frac{5}{11}a^{3}+\frac{3}{11}a^{2}-\frac{5}{11}a-\frac{4}{11}$, $\frac{1}{473}a^{12}-\frac{21}{473}a^{11}+\frac{225}{473}a^{10}+\frac{157}{473}a^{9}+\frac{236}{473}a^{8}-\frac{2}{43}a^{7}+\frac{36}{473}a^{6}+\frac{96}{473}a^{5}-\frac{109}{473}a^{4}-\frac{178}{473}a^{3}-\frac{3}{11}a^{2}+\frac{199}{473}a-\frac{161}{473}$, $\frac{1}{13547955806627}a^{13}+\frac{12379551315}{13547955806627}a^{12}-\frac{137438584253}{13547955806627}a^{11}-\frac{4055119528776}{13547955806627}a^{10}+\frac{4455941572171}{13547955806627}a^{9}-\frac{3294739031500}{13547955806627}a^{8}-\frac{3017158066746}{13547955806627}a^{7}+\frac{412587467959}{1231632346057}a^{6}-\frac{4094210723176}{13547955806627}a^{5}+\frac{5407855341135}{13547955806627}a^{4}-\frac{4000871983637}{13547955806627}a^{3}+\frac{2163961037325}{13547955806627}a^{2}-\frac{258628756101}{13547955806627}a+\frac{584840440497}{13547955806627}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{249214540611}{13547955806627}a^{13}-\frac{156402356573}{13547955806627}a^{12}-\frac{3855707199669}{13547955806627}a^{11}+\frac{2900799658888}{13547955806627}a^{10}+\frac{16769529888435}{13547955806627}a^{9}-\frac{1168150292015}{13547955806627}a^{8}-\frac{31475389875043}{13547955806627}a^{7}-\frac{114195498259558}{13547955806627}a^{6}+\frac{113556649365682}{13547955806627}a^{5}+\frac{155258631781846}{13547955806627}a^{4}+\frac{302284720470060}{13547955806627}a^{3}-\frac{30789252160441}{13547955806627}a^{2}+\frac{41946477403143}{13547955806627}a+\frac{1170416907525}{13547955806627}$, $\frac{171171261033}{13547955806627}a^{13}+\frac{9786836236}{13547955806627}a^{12}-\frac{2649087916537}{13547955806627}a^{11}+\frac{76326058042}{13547955806627}a^{10}+\frac{11605557658300}{13547955806627}a^{9}+\frac{8900974043059}{13547955806627}a^{8}-\frac{15525888325062}{13547955806627}a^{7}-\frac{100203465601103}{13547955806627}a^{6}+\frac{8079453646958}{13547955806627}a^{5}+\frac{12846624071306}{1231632346057}a^{4}+\frac{340763708794609}{13547955806627}a^{3}+\frac{193123852199292}{13547955806627}a^{2}+\frac{1051966878287}{1231632346057}a+\frac{694327100451}{13547955806627}$, $\frac{3980419701685}{13547955806627}a^{13}-\frac{348424679334}{1231632346057}a^{12}-\frac{59968476603489}{13547955806627}a^{11}+\frac{65682364962404}{13547955806627}a^{10}+\frac{238977054232750}{13547955806627}a^{9}-\frac{86685464716791}{13547955806627}a^{8}-\frac{433966640482207}{13547955806627}a^{7}-\frac{17\!\cdots\!20}{13547955806627}a^{6}+\frac{23\!\cdots\!43}{13547955806627}a^{5}+\frac{16\!\cdots\!63}{13547955806627}a^{4}+\frac{45\!\cdots\!08}{13547955806627}a^{3}-\frac{14\!\cdots\!83}{13547955806627}a^{2}+\frac{874655056726277}{13547955806627}a+\frac{77682784133305}{13547955806627}$, $\frac{19462070487}{13547955806627}a^{13}+\frac{99421249347}{13547955806627}a^{12}+\frac{119540249788}{13547955806627}a^{11}-\frac{2840530433364}{13547955806627}a^{10}-\frac{3716825637179}{13547955806627}a^{9}+\frac{29438971644617}{13547955806627}a^{8}+\frac{505791729035}{1231632346057}a^{7}-\frac{96032986041455}{13547955806627}a^{6}-\frac{12420017447137}{13547955806627}a^{5}-\frac{42463057575091}{13547955806627}a^{4}+\frac{560364290938929}{13547955806627}a^{3}-\frac{190913550409546}{13547955806627}a^{2}+\frac{81338060676839}{13547955806627}a+\frac{10836939710757}{13547955806627}$, $\frac{10653201712321}{13547955806627}a^{13}-\frac{11322117245140}{13547955806627}a^{12}-\frac{3701685488233}{315068739689}a^{11}+\frac{4446076227163}{315068739689}a^{10}+\frac{617851254831488}{13547955806627}a^{9}-\frac{285311939817199}{13547955806627}a^{8}-\frac{11\!\cdots\!70}{13547955806627}a^{7}-\frac{45\!\cdots\!57}{13547955806627}a^{6}+\frac{66\!\cdots\!42}{13547955806627}a^{5}+\frac{37\!\cdots\!10}{13547955806627}a^{4}+\frac{12\!\cdots\!38}{13547955806627}a^{3}-\frac{51\!\cdots\!22}{13547955806627}a^{2}+\frac{29\!\cdots\!43}{13547955806627}a-\frac{88774256381765}{13547955806627}$, $\frac{3560896677709}{13547955806627}a^{13}-\frac{2040197050017}{13547955806627}a^{12}-\frac{55433524138309}{13547955806627}a^{11}+\frac{37550160608573}{13547955806627}a^{10}+\frac{243355571059108}{13547955806627}a^{9}+\frac{11008281649182}{13547955806627}a^{8}-\frac{446332032931366}{13547955806627}a^{7}-\frac{17\!\cdots\!14}{13547955806627}a^{6}+\frac{14\!\cdots\!66}{13547955806627}a^{5}+\frac{26\!\cdots\!19}{13547955806627}a^{4}+\frac{47\!\cdots\!84}{13547955806627}a^{3}-\frac{101951615464299}{13547955806627}a^{2}-\frac{12\!\cdots\!70}{13547955806627}a-\frac{56733543573087}{13547955806627}$ | 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: | \( 421014.85114896303 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 421014.85114896303 \cdot 1}{2\cdot\sqrt{1977326743000000000000}}\cr\approx \mathstrut & 1.83015249588756 \end{aligned}\]
Galois group
A solvable group of order 42 |
The 7 conjugacy class representatives for $F_7$ |
Character table for $F_7$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), 7.1.16807000000.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 42 |
Degree 7 sibling: | 7.1.16807000000.2 |
Degree 21 sibling: | deg 21 |
Minimal sibling: | 7.1.16807000000.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | R | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.1.0.1}{1} }^{14}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
\(5\) | 5.14.12.1 | $x^{14} + 28 x^{13} + 350 x^{12} + 2576 x^{11} + 12404 x^{10} + 41104 x^{9} + 96152 x^{8} + 160650 x^{7} + 192444 x^{6} + 165676 x^{5} + 106232 x^{4} + 65016 x^{3} + 59920 x^{2} + 57232 x + 27193$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |