Minimal Weierstrass equation
Minimal equation
Minimal equation
Simplified equation
|
\(y^2+xy=x^3+x^2-2x\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3+x^2z-2xz^2\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-3267x+45630\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(2, 2\right)\)
|
| $\hat{h}(P)$ | ≈ | $0.14325389294088007147627441303$ |
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
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 102 \) | = | $2 \cdot 3 \cdot 17$ |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | $612 $ | = | $2^{2} \cdot 3^{2} \cdot 17 $ |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
| j-invariant: | \( \frac{1771561}{612} \) | = | $2^{-2} \cdot 3^{-2} \cdot 11^{6} \cdot 17^{-1}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $-0.76352819056830652556218343290\dots$ | ||
| Stable Faltings height: | $-0.76352819056830652556218343290\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
| |||
| Analytic rank: | $1$ | ||
sage: E.regulator()
magma: Regulator(E);
| |||
| Regulator: | $0.14325389294088007147627441303\dots$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
| |||
| Real period: | $4.7278638235414655119977559586\dots$ | ||
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: | $ 4 $ = $ 2\cdot2\cdot1 $ | ||
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) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
| |||
| Special value: | $ L'(E,1) $ ≈ $ 0.67728489801666901020123734615 $ | ||
Modular invariants
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 8 | ||
| $ \Gamma_0(N) $-optimal: | yes | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is semistable. There are 3 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2B | 8.6.0.4 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ 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 | ord | ord | ss | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-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 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.2.17.1-612.1-a2 |
| $4$ | 4.0.1088.1 | \(\Z/4\Z\) | Not in database |
| $8$ | 8.4.2002066523136.1 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | 8.0.342102016.4 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | 8.2.236727913392.1 | \(\Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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.
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 $\log a_n \sim \frac12 \hat h(P) n^2$.