Minimal Weierstrass equation
\(y^2+xy+y=x^3-x^2-23x-424542\)
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
| \(P\) | = | \( \left(146, 1560\right) \) |
| \(\hat{h}(P)\) | ≈ | $0.96593385048737423590699481332$ |
Integral points
\( \left(89, 477\right) \), \( \left(89, -567\right) \), \( \left(146, 1560\right) \), \( \left(146, -1707\right) \)
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: | \(-77863895318979 \) | = | \(-1 \cdot 3^{9} \cdot 7 \cdot 11^{7} \cdot 29 \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{1}{60291} \) | = | \(-1 \cdot 3^{-3} \cdot 7^{-1} \cdot 11^{-1} \cdot 29^{-1}\) |
| 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: | \(1\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(0.96593385048737423590699481332\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.28002610291477947673209746451\) | ||
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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: | \( 16 \) = \( 2^{2}\cdot1\cdot2^{2}\cdot1 \) | ||
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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: | \(1\) (exact) | ||
Modular invariants
Modular form 221067.2.a.f
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: | 414720 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 4.3277870692071467168467914091351050277 \)
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(3\) | \(4\) | \(I_3^{*}\) | Additive | -1 | 2 | 9 | 3 |
| \(7\) | \(1\) | \(I_{1}\) | Split multiplicative | -1 | 1 | 1 | 1 |
| \(11\) | \(4\) | \(I_1^{*}\) | Additive | -1 | 2 | 7 | 1 |
| \(29\) | \(1\) | \(I_{1}\) | 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 \( p \) .
$p$-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 no rational isogenies. Its isogeny class 221067.f consists of this curve only.
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 |
|---|---|---|---|
| $3$ | 3.1.6699.1 | \(\Z/2\Z\) | Not in database |
| $6$ | 6.0.300628350099.1 | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/3\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/4\Z\) | 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.