magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![414212142950309582401202559, 43071240857705358819374952, 666759835809172158276111, -40091843709134882829360, -708298583106976109136, 12030481855773499104, 229281249311034354, -1435876427859264, -32146732228590, 73963665216, 2221160193, -1377936, -75054, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 75054*x^12 - 1377936*x^11 + 2221160193*x^10 + 73963665216*x^9 - 32146732228590*x^8 - 1435876427859264*x^7 + 229281249311034354*x^6 + 12030481855773499104*x^5 - 708298583106976109136*x^4 - 40091843709134882829360*x^3 + 666759835809172158276111*x^2 + 43071240857705358819374952*x + 414212142950309582401202559)
gp: K = bnfinit(x^14 - 75054*x^12 - 1377936*x^11 + 2221160193*x^10 + 73963665216*x^9 - 32146732228590*x^8 - 1435876427859264*x^7 + 229281249311034354*x^6 + 12030481855773499104*x^5 - 708298583106976109136*x^4 - 40091843709134882829360*x^3 + 666759835809172158276111*x^2 + 43071240857705358819374952*x + 414212142950309582401202559, 1)
Normalized defining polynomial
\( x^{14} - 75054 x^{12} - 1377936 x^{11} + 2221160193 x^{10} + 73963665216 x^{9} - 32146732228590 x^{8} - 1435876427859264 x^{7} + 229281249311034354 x^{6} + 12030481855773499104 x^{5} - 708298583106976109136 x^{4} - 40091843709134882829360 x^{3} + 666759835809172158276111 x^{2} + 43071240857705358819374952 x + 414212142950309582401202559 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $14$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[10, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(103318334065426770765470197661413735479481944901632942974418171604331965573259563498841619061291062477630041009194956313095225461332708654891833432491165229458866561911760649155550195068503373410096128256000000=2^{14}\cdot 3^{22}\cdot 5^{6}\cdot 7^{14}\cdot 23^{2}\cdot 211^{2}\cdot 68389^{6}\cdot 119737^{2}\cdot 67580323^{6}\cdot 332926921^{6}\cdot 65049457057236257^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $850{,}323{,}339{,}436{,}883.68$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 3, 5, 7, 23, 211, 68389, 119737, 67580323, 332926921, 65049457057236257$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12}$, $\frac{1}{75} a^{13} + \frac{11}{75} a^{12} - \frac{11}{25} a^{11} - \frac{8}{25} a^{10} - \frac{7}{25} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{25} a^{6} - \frac{2}{5} a^{5} + \frac{8}{25} a^{4} + \frac{1}{25} a^{3} - \frac{9}{25} a^{2} - \frac{12}{25} a + \frac{2}{25}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
Not computed
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $11$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A non-solvable group of order 43589145600 |
| The 72 conjugacy class representatives for A14 are not computed |
| Character table for A14 is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 | R | R | ${\href{/LocalNumberField/11.13.0.1}{13} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.11.0.1}{11} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ | ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.13.0.1}{13} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.13.0.1}{13} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.12.12.21 | $x^{12} + 44 x^{10} + 45 x^{8} - 48 x^{6} + 59 x^{4} - 60 x^{2} + 23$ | $2$ | $6$ | $12$ | 12T134 | $[2, 2, 2, 2, 2, 2]^{6}$ | |
| $3$ | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.12.21.52 | $x^{12} + 12 x^{11} + 3 x^{10} + 3 x^{9} + 9 x^{8} - 9 x^{7} - 9 x^{6} + 9 x^{5} + 9 x^{4} - 3 x^{3} + 9 x^{2} + 9 x + 6$ | $12$ | $1$ | $21$ | 12T169 | $[3/2, 2, 9/4, 9/4]_{4}^{2}$ | |
| $5$ | $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $7$ | 7.7.7.6 | $x^{7} + 28 x + 7$ | $7$ | $1$ | $7$ | $F_7$ | $[7/6]_{6}$ |
| 7.7.7.5 | $x^{7} + 7 x + 7$ | $7$ | $1$ | $7$ | $F_7$ | $[7/6]_{6}$ | |
| $23$ | 23.3.2.1 | $x^{3} - 23$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.7.0.1 | $x^{7} - x + 8$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 211 | Data not computed | ||||||
| 68389 | Data not computed | ||||||
| 119737 | Data not computed | ||||||
| 67580323 | Data not computed | ||||||
| 332926921 | Data not computed | ||||||
| 65049457057236257 | Data not computed | ||||||