# Number fields downloaded from the LMFDB on 15 October 2026.
# Search link: https://www.lmfdb.org/NumberField/?galois_group=11T3
# Query "{'degree': 11, 'galois_label': '11T3'}" returned 3 fields, sorted by degree.

# Each entry in the following data list has the form:
#    [Label, Polynomial, Discriminant, Galois group, Class group]
# For more details, see the definitions at the bottom of the file.



"11.11.1771197285652216321.1"	[-19, -67, 197, 516, -13, -449, -53, 151, 14, -21, -1, 1]	1771197285652216321	"11T3"	[]
"11.11.123610132462587890625.1"	[4, 11, -80, -15, 240, -63, -207, 105, 30, -20, -1, 1]	123610132462587890625	"11T3"	[]
"11.11.2713285598714072534889.1"	[-837, -2673, 0, 4455, 0, -2079, 0, 396, 0, -33, 0, 1]	2713285598714072534889	"11T3"	[]
"11.11.25132452704633039250241.1"	[64, -1319, 5301, -5667, -1592, 5783, -2975, 164, 245, -49, -3, 1]	25132452704633039250241	"11T3"	[]
"11.11.28501271562015485137921.1"	[97441, -47911, -92821, 32219, 27863, -8260, -3574, 977, 200, -52, -4, 1]	28501271562015485137921	"11T3"	[]
"11.11.448727830698946884765625.1"	[-12675, -34375, 0, 34375, 0, -9625, 0, 1100, 0, -55, 0, 1]	448727830698946884765625	"11T3"	[]
"11.11.143870245619672894373619681.1"	[356992, 1268768, -299504, -657992, 208536, 76430, -34679, -269, 1534, -165, -5, 1]	143870245619672894373619681	"11T3"	[]
"11.11.933299499650312459853148801.1"	[83031, 4197766, -2482765, -753338, 682397, -26840, -51764, 8054, 712, -171, -2, 1]	933299499650312459853148801	"11T3"	[]
"11.11.6578516425613671512252024001.1"	[5322020, 2691611, -5613883, -5687237, -1600840, 90245, 117915, 15324, -873, -248, -2, 1]	6578516425613671512252024001	"11T3"	[]
"11.11.27041735968924735397910759841.1"	[-12691, -261279, -171775, 1031588, -23214, -187211, 3981, 9850, -36, -178, 0, 1]	27041735968924735397910759841	"11T3"	[]
"11.11.97011105196856107941421953601.1"	[-239221, -2549470, -5329883, -3128188, 308993, 869434, 292220, 32709, -839, -343, -5, 1]	97011105196856107941421953601	"11T3"	[]


# Label --
#    Each (global) number field has a unique label of the form d.r.D.i where
#    <ul>
#    <li>\(d\) is the degree;
#    <li>\(r\) is the real signature;  the full signature is therefore \([r,(d-r)/2]\);
#    <li>\(D\) is the absolute value of the discriminant;
#    <li>\(i\) is the index, counting from 1.  This is in case there is more than one
#      field with the same signature and absolute value of the
#      discriminant: for example <a href="/NumberField/4.0.1008.1">4.0.1008.1</a> and <a href="/NumberField/4.0.1008.2">4.0.1008.2</a>.
#    </ul>
#    The discriminant portion of the label can take the form \(a_1\) e \(\epsilon_1\) _ \(a_2\) e \(\epsilon_2\) _ \(\cdots\) _ \(a_k\) e \(\epsilon_k\) to mean the absolute value of the
#    discriminant equals \(a_1^{\epsilon_1}a_2^{\epsilon_2}\cdots a_k^{\epsilon_k}\).  The separators are the letter e and the underscore symbol.


#Polynomial (coeffs) --
#    A **defining polynomial** of a number field $K$ is an irreducible polynomial $f\in\Q[x]$ such that $K\cong \mathbb{Q}(a)$, where $a$ is a root of $f(x)$. Equivalently, it is a polynomial $f\in \Q[x]$ such that $K \cong \Q[x]/(f)$.

#    A root \(a \in K\) of the defining polynomial is a generator of \(K\).




#Discriminant (disc) --
#    The **discriminant** of a number field $K$ is the square of the determinant of the matrix
#    \[
#    \left( \begin{array}{ccc}
#     \sigma_1(\beta_1) & \cdots & \sigma_1(\beta_n) \\
#    \vdots & & \vdots \\
#    \sigma_n(\beta_1) & \cdots & \sigma_n(\beta_n) \\
#    \end{array} \right)
#    \]
#    where $\sigma_1,..., \sigma_n$ are the embeddings of $K$ into the complex numbers $\mathbb{C}$, and $\{\beta_1, \ldots, \beta_n\}$ is an integral basis for the ring of integers of $K$.

#    The discriminant of $K$ is a non-zero integer divisible exactly by the primes which ramify in $K$.



#Galois group (galois_label) --
#    Let $K$ be a finite degree $n$ separable extension of a field $F$, and $K^{gal}$ be its
#    Galois (or normal) closure.
#    The **Galois group** for $K/F$ is the automorphism group $\Aut(K^{gal}/F)$.

#    This automorphism group acts on the $n$ embeddings $K\hookrightarrow K^{gal}$ via composition.  As a result, we get an injection $\Aut(K^{gal}/F)\hookrightarrow S_n$, which is well-defined up to the labelling of the $n$ embeddings, which corresponds to being well-defined up to conjugation in $S_n$.

#    We use the notation $\Gal(K/F)$ for $\Aut(K/F)$ when $K=K^{gal}$.

#    There is a naming convention for Galois groups up to degree $47$.





#Class group (class_group) --
#    The **ideal class group** of a number field $K$ with ring of integers $O_K$ is the group of equivalence classes of ideals, given by the quotient of the multiplicative group of all fractional ideals of $O_K$ by the subgroup of principal fractional ideals.

#    Since $K$ is a number field, the ideal class group of $K$ is a finite abelian group, and so has the structure of a product of cyclic groups encoded by a finite list $[a_1,\dots,a_n]$, where the $a_i$ are positive integers with $a_i\mid a_{i+1}$ for $1\leq i < n$.