Minimal Weierstrass equation
\(y^2+y=x^3+562650x-162148077\)
Mordell-Weil group structure
trivial
Integral points
None
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 221067 \) | = | \(3^{2} \cdot 7 \cdot 11^{2} \cdot 29\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-22757883409104720723 \) | = | \(-1 \cdot 3^{6} \cdot 7^{3} \cdot 11^{12} \cdot 29 \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{15252992000000}{17621717267} \) | = | \(2^{15} \cdot 5^{6} \cdot 7^{-3} \cdot 11^{-6} \cdot 29^{-1} \cdot 31^{3}\) |
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
BSD invariants
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sage: E.rank()
magma: Rank(E);
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| Analytic rank: | \(0\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(1\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.11518371783903265608337716991\) | ||
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sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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| Tamagawa product: | \( 2 \) = \( 1\cdot1\cdot2\cdot1 \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(1\) | ||
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sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | \(9\) = $3^2$ (exact) | ||
Modular invariants
Modular form 221067.2.a.q
For more coefficients, see the Downloads section to the right.
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sage: E.modular_degree()
magma: ModularDegree(E);
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| Modular degree: | 3628800 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L(E,1) \) ≈ \( 2.0733069211025878095007890584192101308 \)
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(3\) | \(1\) | \(I_0^{*}\) | Additive | -1 | 2 | 6 | 0 |
| \(7\) | \(1\) | \(I_{3}\) | Non-split multiplicative | 1 | 1 | 3 | 3 |
| \(11\) | \(2\) | \(I_6^{*}\) | Additive | -1 | 2 | 12 | 6 |
| \(29\) | \(1\) | \(I_{1}\) | Split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The 2-adic representation attached to this elliptic curve is 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 |
|---|---|
| \(3\) | B |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
No Iwasawa invariant data is available for this curve.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class 221067.q
consists of 2 curves linked by isogenies of
degree 3.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-11}) \) | \(\Z/3\Z\) | Not in database |
| $3$ | 3.1.812.1 | \(\Z/2\Z\) | Not in database |
| $6$ | 6.0.133846832.1 | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $6$ | 6.2.2058822141057.2 | \(\Z/3\Z\) | Not in database |
| $6$ | 6.0.877586864.2 | \(\Z/6\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/4\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/3\Z \times \Z/3\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $18$ | 18.0.73519638586885947190079122857733741544492536473819.1 | \(\Z/9\Z\) | Not in database |
| $18$ | 18.2.3536719966069155332375734542253987838137016881152.1 | \(\Z/6\Z\) | 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.