Minimal Weierstrass equation
magma: E := EllipticCurve("33a2");
sage: E = EllipticCurve("33a2")
gp: E = ellinit("33a2")
\( y^2 + x y = x^{3} + x^{2} - 6 x - 9 \)
Mordell-Weil group structure
Torsion generators
\( \left(-2, 1\right) \)
Integral points
\( \left(-2, 1\right) \)
Invariants
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magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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| \( N \) | = | \( 33 \) | = | \(3 \cdot 11\) | |
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magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
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| \(\Delta\) | = | \(297 \) | = | \(3^{3} \cdot 11 \) | |
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magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
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| \(j \) | = | \( \frac{30664297}{297} \) | = | \(3^{-3} \cdot 11^{-1} \cdot 313^{3}\) | |
| \( \text{End} (E) \) | = | \(\Z\) | (no Complex Multiplication) | ||
| \( \text{ST} (E) \) | = | $\mathrm{SU}(2)$ | |||
BSD invariants
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magma: Rank(E);
sage: E.rank()
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| \( r \) | = | \(0\) | |
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magma: Regulator(E);
sage: E.regulator()
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| \( \text{Reg} \) | = | \(1\) | |
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magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
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| \( \Omega \) | ≈ | \(2.98935659097\) | |
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magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
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| \( \prod_p c_p \) | = | \( 1 \) = \( 1\cdot1 \) | |
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magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
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| \( \#E_{\text{tor}} \) | = | \(2\) | |
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magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
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| Ш\(_{\text{an}} \) | = | \(1\) (exact) | |
Modular invariants
Modular form 33.2.1.a
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
Modular degree and optimality
Special L-value attached to the curve
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar[2]/factorial(ar[1])
\( L(E,1) \) ≈ \( 0.747339147743 \)
Local data
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(3\) | \(1\) | \( I_{3} \) | Non-split multiplicative | 1 | 1 | 3 | 3 |
| \(11\) | \(1\) | \( I_{1} \) | Split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X13d.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 4 & 1 \end{array}\right)$ and has index 12.
sage: [rho.image_type(p) for p in rho.non_surjective()]
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(2\) | B |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
Iwasawa invariants
| $p$ | 2 | 3 | 11 |
|---|---|---|---|
| Reduction type | ordinary | nonsplit | split |
| $\lambda$-invariant(s) | 0 | 0 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class 33.a
consists of 4 curves linked by isogenies of
degrees dividing 4.
Growth of torsion in number fields
The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base-change curve |
|---|---|---|---|
| 2 | \(\Q(\sqrt{-11}) \) | \(\Z/4\Z\) | 2.0.11.1-99.2-a3 |
| \(\Q(\sqrt{33}) \) | \(\Z/2\Z \times \Z/2\Z\) | 2.2.33.1-33.1-a4 | |
| \(\Q(\sqrt{-3}) \) | \(\Z/4\Z\) | 2.0.3.1-363.1-a3 | |
| 4 | 4.0.3993.1 | \(\Z/8\Z\) | Not in database |
| \(\Q(\sqrt{-3}, \sqrt{-11})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.