Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^19 + 4*x^16 - 4*x^15 + x^14 - 6*x^13 + 7*x^12 - 3*x^11 + 12*x^10 + 2*x^9 + 18*x^8 + 4*x^7 + 13*x^6 + 5*x^5 + 6*x^4 + x^3 + 4*x^2 - x + 1)
gp: K = bnfinit(y^22 - 4*y^19 + 4*y^16 - 4*y^15 + y^14 - 6*y^13 + 7*y^12 - 3*y^11 + 12*y^10 + 2*y^9 + 18*y^8 + 4*y^7 + 13*y^6 + 5*y^5 + 6*y^4 + y^3 + 4*y^2 - y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 4*x^19 + 4*x^16 - 4*x^15 + x^14 - 6*x^13 + 7*x^12 - 3*x^11 + 12*x^10 + 2*x^9 + 18*x^8 + 4*x^7 + 13*x^6 + 5*x^5 + 6*x^4 + x^3 + 4*x^2 - x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 4*x^19 + 4*x^16 - 4*x^15 + x^14 - 6*x^13 + 7*x^12 - 3*x^11 + 12*x^10 + 2*x^9 + 18*x^8 + 4*x^7 + 13*x^6 + 5*x^5 + 6*x^4 + x^3 + 4*x^2 - x + 1)
\( x^{22} - 4 x^{19} + 4 x^{16} - 4 x^{15} + x^{14} - 6 x^{13} + 7 x^{12} - 3 x^{11} + 12 x^{10} + 2 x^{9} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-110395262036162513388217347\)
\(\medspace = -\,3^{11}\cdot 971^{2}\cdot 25709231^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.27\) | 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: | $3^{1/2}971^{1/2}25709231^{1/2}\approx 273662.18208404316$ | ||
| Ramified primes: |
\(3\), \(971\), \(25709231\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{3835646}a^{21}+\frac{160649}{3835646}a^{20}-\frac{21455}{1917823}a^{19}-\frac{396366}{1917823}a^{18}+\frac{1433247}{3835646}a^{17}+\frac{810696}{1917823}a^{16}-\frac{1798625}{3835646}a^{15}-\frac{423157}{3835646}a^{14}-\frac{297417}{1917823}a^{13}-\frac{120651}{3835646}a^{12}+\frac{974569}{3835646}a^{11}-\frac{1780973}{3835646}a^{10}-\frac{553605}{1917823}a^{9}+\frac{150493}{3835646}a^{8}+\frac{473237}{3835646}a^{7}-\frac{1288549}{3835646}a^{6}-\frac{47137}{3835646}a^{5}-\frac{473352}{1917823}a^{4}+\frac{74328}{1917823}a^{3}+\frac{705749}{3835646}a^{2}+\frac{10991}{3835646}a-\frac{621825}{3835646}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $10$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{2446799}{3835646} a^{21} - \frac{364147}{1917823} a^{20} + \frac{924955}{3835646} a^{19} + \frac{9956951}{3835646} a^{18} + \frac{2441111}{3835646} a^{17} - \frac{1997558}{1917823} a^{16} - \frac{5101796}{1917823} a^{15} + \frac{8857079}{3835646} a^{14} + \frac{5279489}{3835646} a^{13} + \frac{9758697}{3835646} a^{12} - \frac{14706767}{3835646} a^{11} - \frac{2136465}{3835646} a^{10} - \frac{9731443}{1917823} a^{9} - \frac{6847511}{1917823} a^{8} - \frac{37853405}{3835646} a^{7} - \frac{10577418}{1917823} a^{6} - \frac{10185582}{1917823} a^{5} - \frac{8712443}{1917823} a^{4} - \frac{6192274}{1917823} a^{3} - \frac{6611313}{3835646} a^{2} - \frac{3403586}{1917823} a + \frac{421412}{1917823} \)
(order $6$)
| 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: |
$\frac{421412}{1917823}a^{21}+\frac{2446799}{3835646}a^{20}+\frac{364147}{1917823}a^{19}-\frac{4296251}{3835646}a^{18}-\frac{9956951}{3835646}a^{17}-\frac{2441111}{3835646}a^{16}+\frac{3683206}{1917823}a^{15}+\frac{3416148}{1917823}a^{14}-\frac{8014255}{3835646}a^{13}-\frac{10336433}{3835646}a^{12}-\frac{3858929}{3835646}a^{11}+\frac{12178295}{3835646}a^{10}+\frac{12250353}{3835646}a^{9}+\frac{10574267}{1917823}a^{8}+\frac{14432927}{1917823}a^{7}+\frac{41224701}{3835646}a^{6}+\frac{16055774}{1917823}a^{5}+\frac{12292642}{1917823}a^{4}+\frac{11240915}{1917823}a^{3}+\frac{6613686}{1917823}a^{2}+\frac{9982609}{3835646}a+\frac{2982174}{1917823}$, $\frac{3638966}{1917823}a^{21}-\frac{6064864}{1917823}a^{20}-\frac{6553692}{1917823}a^{19}-\frac{15987078}{1917823}a^{18}+\frac{27043217}{1917823}a^{17}+\frac{28997225}{1917823}a^{16}+\frac{20701765}{1917823}a^{15}-\frac{51109369}{1917823}a^{14}-\frac{10594464}{1917823}a^{13}-\frac{9174414}{1917823}a^{12}+\frac{77219558}{1917823}a^{11}-\frac{10519133}{1917823}a^{10}+\frac{18357203}{1917823}a^{9}-\frac{77913801}{1917823}a^{8}-\frac{19645554}{1917823}a^{7}-\frac{113972395}{1917823}a^{6}-\frac{61221558}{1917823}a^{5}-\frac{74651801}{1917823}a^{4}-\frac{42642551}{1917823}a^{3}-\frac{36876686}{1917823}a^{2}+\frac{1594464}{1917823}a-\frac{14374294}{1917823}$, $\frac{3680737}{3835646}a^{21}-\frac{19482923}{3835646}a^{20}-\frac{7685956}{1917823}a^{19}-\frac{8031766}{1917823}a^{18}+\frac{84057753}{3835646}a^{17}+\frac{33299836}{1917823}a^{16}+\frac{21052915}{3835646}a^{15}-\frac{119267453}{3835646}a^{14}-\frac{1209699}{1917823}a^{13}+\frac{10329739}{3835646}a^{12}+\frac{166450117}{3835646}a^{11}-\frac{51462429}{3835646}a^{10}-\frac{2811969}{1917823}a^{9}-\frac{237473801}{3835646}a^{8}-\frac{143722983}{3835646}a^{7}-\frac{367402815}{3835646}a^{6}-\frac{227427565}{3835646}a^{5}-\frac{124218109}{1917823}a^{4}-\frac{78097603}{1917823}a^{3}-\frac{118358929}{3835646}a^{2}-\frac{14920579}{3835646}a-\frac{48152471}{3835646}$, $\frac{3780282}{1917823}a^{21}-\frac{15876731}{3835646}a^{20}-\frac{8184749}{1917823}a^{19}-\frac{33732805}{3835646}a^{18}+\frac{70238885}{3835646}a^{17}+\frac{72548397}{3835646}a^{16}+\frac{22964747}{1917823}a^{15}-\frac{61831956}{1917823}a^{14}-\frac{25865659}{3835646}a^{13}-\frac{17371497}{3835646}a^{12}+\frac{186017163}{3835646}a^{11}-\frac{23421153}{3835646}a^{10}+\frac{32998773}{3835646}a^{9}-\frac{104433782}{1917823}a^{8}-\frac{41187548}{1917823}a^{7}-\frac{304840093}{3835646}a^{6}-\frac{84848447}{1917823}a^{5}-\frac{102426953}{1917823}a^{4}-\frac{58346981}{1917823}a^{3}-\frac{49225232}{1917823}a^{2}+\frac{806157}{3835646}a-\frac{19299603}{1917823}$, $\frac{6176957}{3835646}a^{21}-\frac{1847213}{3835646}a^{20}-\frac{1207489}{1917823}a^{19}-\frac{12211294}{1917823}a^{18}+\frac{8364673}{3835646}a^{17}+\frac{4778949}{1917823}a^{16}+\frac{22917317}{3835646}a^{15}-\frac{36969133}{3835646}a^{14}+\frac{2412852}{1917823}a^{13}-\frac{24542021}{3835646}a^{12}+\frac{58383939}{3835646}a^{11}-\frac{23374313}{3835646}a^{10}+\frac{27965740}{1917823}a^{9}-\frac{6867821}{3835646}a^{8}+\frac{91660483}{3835646}a^{7}-\frac{8778545}{3835646}a^{6}+\frac{39022211}{3835646}a^{5}-\frac{2278347}{1917823}a^{4}+\frac{3622811}{1917823}a^{3}-\frac{18399861}{3835646}a^{2}+\frac{11507125}{3835646}a-\frac{16409877}{3835646}$, $\frac{3124514}{1917823}a^{21}-\frac{28460107}{3835646}a^{20}-\frac{11314571}{1917823}a^{19}-\frac{27381213}{3835646}a^{18}+\frac{123113987}{3835646}a^{17}+\frac{98247375}{3835646}a^{16}+\frac{17607048}{1917823}a^{15}-\frac{89507418}{1917823}a^{14}-\frac{4029283}{3835646}a^{13}+\frac{11867877}{3835646}a^{12}+\frac{251896909}{3835646}a^{11}-\frac{79438519}{3835646}a^{10}-\frac{637913}{3835646}a^{9}-\frac{172715277}{1917823}a^{8}-\frac{96038863}{1917823}a^{7}-\frac{534533897}{3835646}a^{6}-\frac{164014088}{1917823}a^{5}-\frac{183270177}{1917823}a^{4}-\frac{112950743}{1917823}a^{3}-\frac{86145564}{1917823}a^{2}-\frac{22942227}{3835646}a-\frac{34985316}{1917823}$, $\frac{5448663}{3835646}a^{21}-\frac{461129}{1917823}a^{20}-\frac{2245223}{3835646}a^{19}-\frac{21981477}{3835646}a^{18}+\frac{4369557}{3835646}a^{17}+\frac{4570751}{1917823}a^{16}+\frac{10993600}{1917823}a^{15}-\frac{29242845}{3835646}a^{14}-\frac{96393}{3835646}a^{13}-\frac{22121195}{3835646}a^{12}+\frac{48907077}{3835646}a^{11}-\frac{13475611}{3835646}a^{10}+\frac{23565028}{1917823}a^{9}-\frac{339422}{1917823}a^{8}+\frac{80292843}{3835646}a^{7}+\frac{1329339}{1917823}a^{6}+\frac{16915660}{1917823}a^{5}-\frac{1130224}{1917823}a^{4}+\frac{1540554}{1917823}a^{3}-\frac{15419837}{3835646}a^{2}+\frac{3033752}{1917823}a-\frac{6981539}{1917823}$, $\frac{27947327}{3835646}a^{21}+\frac{95094}{1917823}a^{20}-\frac{6375481}{1917823}a^{19}-\frac{120090815}{3835646}a^{18}-\frac{188147}{1917823}a^{17}+\frac{54143021}{3835646}a^{16}+\frac{146933825}{3835646}a^{15}-\frac{112345767}{3835646}a^{14}-\frac{37117799}{3835646}a^{13}-\frac{78961346}{1917823}a^{12}+\frac{107594663}{1917823}a^{11}+\frac{15861}{1917823}a^{10}+\frac{293627685}{3835646}a^{9}+\frac{33708521}{3835646}a^{8}+\frac{355751747}{3835646}a^{7}-\frac{1357453}{1917823}a^{6}+\frac{126042219}{3835646}a^{5}-\frac{8297041}{1917823}a^{4}+\frac{1952682}{1917823}a^{3}-\frac{79410099}{3835646}a^{2}+\frac{17735538}{1917823}a-\frac{43360673}{3835646}$, $\frac{18263189}{1917823}a^{21}+\frac{6464810}{1917823}a^{20}-\frac{7951053}{3835646}a^{19}-\frac{78377667}{1917823}a^{18}-\frac{54663773}{3835646}a^{17}+\frac{34883969}{3835646}a^{16}+\frac{192445199}{3835646}a^{15}-\frac{40815723}{1917823}a^{14}-\frac{30532604}{1917823}a^{13}-\frac{233630837}{3835646}a^{12}+\frac{197220297}{3835646}a^{11}+\frac{49729445}{3835646}a^{10}+\frac{413861223}{3835646}a^{9}+\frac{186223789}{3835646}a^{8}+\frac{285506549}{1917823}a^{7}+\frac{122243604}{1917823}a^{6}+\frac{344187011}{3835646}a^{5}+\frac{82463445}{1917823}a^{4}+\frac{64303007}{1917823}a^{3}-\frac{4567565}{1917823}a^{2}+\frac{40122464}{1917823}a-\frac{20496851}{3835646}$, $\frac{4042825}{1917823}a^{21}-\frac{6472463}{1917823}a^{20}-\frac{11471685}{3835646}a^{19}-\frac{16157361}{1917823}a^{18}+\frac{56071467}{3835646}a^{17}+\frac{48281135}{3835646}a^{16}+\frac{31634333}{3835646}a^{15}-\frac{51582525}{1917823}a^{14}+\frac{2192694}{1917823}a^{13}-\frac{12321855}{3835646}a^{12}+\frac{147487547}{3835646}a^{11}-\frac{53079385}{3835646}a^{10}+\frac{44406809}{3835646}a^{9}-\frac{134283961}{3835646}a^{8}+\frac{1821017}{1917823}a^{7}-\frac{106729759}{1917823}a^{6}-\frac{102128233}{3835646}a^{5}-\frac{74307080}{1917823}a^{4}-\frac{45945885}{1917823}a^{3}-\frac{42563378}{1917823}a^{2}+\frac{648488}{1917823}a-\frac{42753229}{3835646}$
| 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: | \( 18267.9467933 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 18267.9467933 \cdot 1}{6\cdot\sqrt{110395262036162513388217347}}\cr\approx \mathstrut & 0.174599028244 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^19 + 4*x^16 - 4*x^15 + x^14 - 6*x^13 + 7*x^12 - 3*x^11 + 12*x^10 + 2*x^9 + 18*x^8 + 4*x^7 + 13*x^6 + 5*x^5 + 6*x^4 + x^3 + 4*x^2 - x + 1)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^22 - 4*x^19 + 4*x^16 - 4*x^15 + x^14 - 6*x^13 + 7*x^12 - 3*x^11 + 12*x^10 + 2*x^9 + 18*x^8 + 4*x^7 + 13*x^6 + 5*x^5 + 6*x^4 + x^3 + 4*x^2 - x + 1, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^22 - 4*x^19 + 4*x^16 - 4*x^15 + x^14 - 6*x^13 + 7*x^12 - 3*x^11 + 12*x^10 + 2*x^9 + 18*x^8 + 4*x^7 + 13*x^6 + 5*x^5 + 6*x^4 + x^3 + 4*x^2 - x + 1);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 4*x^19 + 4*x^16 - 4*x^15 + x^14 - 6*x^13 + 7*x^12 - 3*x^11 + 12*x^10 + 2*x^9 + 18*x^8 + 4*x^7 + 13*x^6 + 5*x^5 + 6*x^4 + x^3 + 4*x^2 - x + 1);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_2\times S_{11}$ (as 22T47):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A non-solvable group of order 79833600 |
| The 112 conjugacy class representatives for $C_2\times S_{11}$ |
| Character table for $C_2\times S_{11}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 11.3.24963663301.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 22 sibling: | data not computed |
| Degree 44 sibling: | 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 | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.6.0.1}{6} }^{2}$ | R | $22$ | ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | $22$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | $18{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.9.0.1}{9} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | $18{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.22.11.2 | $x^{22} + 33 x^{20} + 495 x^{18} + 4455 x^{16} + 26730 x^{14} + 4 x^{13} + 112266 x^{12} - 406 x^{11} + 336798 x^{10} + 1650 x^{9} + 721710 x^{8} + 20196 x^{7} + 1082565 x^{6} - 35640 x^{5} + 1082569 x^{4} - 37422 x^{3} + 649567 x^{2} + 20898 x + 177172$ | $2$ | $11$ | $11$ | 22T1 | $[\ ]_{2}^{11}$ |
|
\(971\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
|
\(25709231\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |