Minimal Weierstrass equation
\(y^2+xy+y=x^3-x^2+636316x+222518926\) 
Mordell-Weil group structure
\(\Z\times \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| \(P\) | = | \(\left(7670, 671647\right)\) |
| \(\hat{h}(P)\) | ≈ | $1.5843770504364586470236343177$ |
Torsion generators
\( \left(-305, 152\right) \)
Integral points
\( \left(-305, 152\right) \), \( \left(7670, 671647\right) \), \( \left(7670, -679318\right) \)
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 221067 \) | = | \(3^{2} \cdot 7 \cdot 11^{2} \cdot 29\) |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | \(-37910116690434740367 \) | = | \(-1 \cdot 3^{6} \cdot 7^{3} \cdot 11^{8} \cdot 29^{4} \) |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
| j-invariant: | \( \frac{22062729659823}{29354283343} \) | = | \(3^{3} \cdot 7^{-3} \cdot 11^{-2} \cdot 29^{-4} \cdot 9349^{3}\) |
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
BSD invariants
|
sage: E.rank()
magma: Rank(E);
|
|||
| Analytic rank: | \(1\) | ||
|
sage: E.regulator()
magma: Regulator(E);
|
|||
| Regulator: | \(1.5843770504364586470236343177\) | ||
|
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
|
|||
| Real period: | \(0.13820012167734371523254121722\) | ||
|
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
|
|||
| Tamagawa product: | \( 96 \) = \( 2\cdot3\cdot2^{2}\cdot2^{2} \) | ||
|
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
|
|||
| Torsion order: | \(2\) | ||
|
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
|||
| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 221067.2.a.l
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 5160960 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 5.2550664276746286120472510040721253935 \)
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(3\) | \(2\) | \(I_0^{*}\) | Additive | -1 | 2 | 6 | 0 |
| \(7\) | \(3\) | \(I_{3}\) | Split multiplicative | -1 | 1 | 3 | 3 |
| \(11\) | \(4\) | \(I_2^{*}\) | Additive | -1 | 2 | 8 | 2 |
| \(29\) | \(4\) | \(I_{4}\) | Split multiplicative | -1 | 1 | 4 | 4 |
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 X13.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 6.
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
\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.
No Iwasawa invariant data is available for this curve.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class 221067.l
consists of 4 curves linked by isogenies of
degrees dividing 4.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/4\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{-231}) \) | \(\Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{33})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | 8.0.35717739450624.29 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | 8.4.41100190149249.1 | \(\Z/8\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/8\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/4\Z \times \Z/4\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/12\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/12\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.