Minimal Weierstrass equation
\( y^2 + x y = x^{3} + x^{2} - 2 x \)
Mordell-Weil group structure
Infinite order Mordell-Weil generator and height
| \(P\) | = | \( \left(-1, 2\right) \) |
| \(\hat{h}(P)\) | ≈ | 0.143253892941 |
Torsion generators
\( \left(0, 0\right) \)
Integral points
\( \left(-2, 2\right) \), \( \left(-2, 0\right) \), \( \left(-1, 2\right) \), \( \left(-1, -1\right) \), \( \left(0, 0\right) \), \( \left(1, 0\right) \), \( \left(1, -1\right) \), \( \left(2, 2\right) \), \( \left(2, -4\right) \), \( \left(8, 20\right) \), \( \left(8, -28\right) \), \( \left(9, 24\right) \), \( \left(9, -33\right) \), \( \left(2738, 141932\right) \), \( \left(2738, -144670\right) \)
Invariants
|
magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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| Conductor: | \( 102 \) | = | \(2 \cdot 3 \cdot 17\) | ||
|
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
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| Discriminant: | \(612 \) | = | \(2^{2} \cdot 3^{2} \cdot 17 \) | ||
|
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
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| j-invariant: | \( \frac{1771561}{612} \) | = | \(2^{-2} \cdot 3^{-2} \cdot 11^{6} \cdot 17^{-1}\) | ||
| Endomorphism ring: | \(\Z\) | (no Complex Multiplication) | |||
| Sato-Tate Group: | $\mathrm{SU}(2)$ | ||||
BSD invariants
|
magma: Rank(E);
sage: E.rank()
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| Rank: | \(1\) | ||
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magma: Regulator(E);
sage: E.regulator()
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| Regulator: | \(0.143253892941\) | ||
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magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
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| Real period: | \(4.72786382354\) | ||
|
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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| Tamagawa product: | \( 4 \) = \( 2\cdot2\cdot1 \) | ||
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magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
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| Torsion order: | \(2\) | ||
|
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 102.2.a.a
|
magma: ModularDegree(E);
sage: E.modular_degree()
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| Modular degree: | 8 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 0.677284898017 \)
Local data
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(2\) | \( I_{2} \) | Non-split multiplicative | 1 | 1 | 2 | 2 |
| \(3\) | \(2\) | \( I_{2} \) | Non-split multiplicative | 1 | 1 | 2 | 2 |
| \(17\) | \(1\) | \( I_{1} \) | Non-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 X15.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 2 & 1 \end{array}\right)$ 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
Note: \(p\)-adic regulator data only exists for primes \(p\ge5\) of good ordinary reduction.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | nonsplit | ordinary | ordinary | ss | ordinary | nonsplit | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary |
| $\lambda$-invariant(s) | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class 102a
consists of 2 curves linked by isogenies of
degree 2.
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 | \(\Q(\sqrt{17}) \) | \(\Z/2\Z \times \Z/2\Z\) | 2.2.17.1-612.1-a2 |
| 4 | 4.0.1088.1 | \(\Z/4\Z\) | 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.
Additional information
This is the elliptic curve $E$ associated to the [Somos-5 sequence] $\{a(n)\}$. Let $T$ be the $2$-torsion point $(0,0)$, and $P$ the point $(2,2)$ such that $E(\Q) = \Z P \oplus \{0, T\}$. Then the $x$- and $y$-coordinates of $nP+T$ have denominators $d_n^2$ and $d_n^3$ where $$d_n = 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217$$ for $1 \leq n \leq 10$, and $d_n = a(n+2)$ in general, satisfying the Somos-5 recurrence $$ d_n d_{n+5} = d_{n+1} d_{n+4} + d_{n+2} d_{n+3}. $$ Thus the regulator of $E$, which is the canonical height $\hat h(P) = 0.143\ldots$, controls the growth of the $a(n)$: asymptotically $\log a_n \sim \frac12 \hat h(P) n^2$.