Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2+3x-1\)
|
(homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z+3xz^2-z^3\)
|
(dehomogenize, simplify) |
\(y^2=x^3+45x-18\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Torsion generators
\( \left(1, 1\right) \)
Integral points
\( \left(1, 1\right) \), \( \left(1, -2\right) \)
Invariants
Conductor: | \( 162 \) | = | $2 \cdot 3^{4}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-1458 $ | = | $-1 \cdot 2 \cdot 3^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{3375}{2} \) | = | $2^{-1} \cdot 3^{3} \cdot 5^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
|
Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $-0.70358699565001647072183612100\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $-1.2528931399840713164194587395\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $2.8006777425532496677500992009\dots$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
|
Tamagawa product: | $ 3 $ = $ 1\cdot3 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
|
Torsion order: | $3$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
Analytic order of Ш: | $1$ (exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
Special value: | $ L(E,1) $ ≈ $ 0.93355924751774988925003306697 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 0.933559248 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 2.800678 \cdot 1.000000 \cdot 3}{3^2} \approx 0.933559248$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 6 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$3$ | $3$ | $IV$ | Additive | -1 | 4 | 6 | 0 |
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 |
---|---|---|
$2$ | 2G | 8.2.0.1 |
$3$ | 3B.1.1 | 3.8.0.1 |
$7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \), index $768$, genus $21$, and generators
$\left(\begin{array}{rr} 281 & 392 \\ 112 & 393 \end{array}\right),\left(\begin{array}{rr} 1 & 426 \\ 42 & 253 \end{array}\right),\left(\begin{array}{rr} 1 & 216 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 462 & 1 \end{array}\right),\left(\begin{array}{rr} 33 & 154 \\ 448 & 129 \end{array}\right),\left(\begin{array}{rr} 85 & 42 \\ 189 & 43 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 168 & 1 \end{array}\right),\left(\begin{array}{rr} 337 & 168 \\ 336 & 169 \end{array}\right),\left(\begin{array}{rr} 295 & 42 \\ 231 & 211 \end{array}\right),\left(\begin{array}{rr} 22 & 321 \\ 189 & 169 \end{array}\right),\left(\begin{array}{rr} 1 & 42 \\ 0 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 420 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 168 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 463 & 282 \\ 420 & 295 \end{array}\right)$.
The torsion field $K:=\Q(E[504])$ is a degree-$15676416$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/504\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 7 and 21.
Its isogeny class 162c
consists of 4 curves linked by isogenies of
degrees dividing 21.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.648.1 | \(\Z/6\Z\) | Not in database |
$6$ | 6.0.3359232.4 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$6$ | 6.0.34992.1 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$6$ | \(\Q(\zeta_{9})\) | \(\Z/21\Z\) | Not in database |
$9$ | 9.3.167365651248.1 | \(\Z/9\Z\) | Not in database |
$12$ | 12.2.51998697814228992.41 | \(\Z/12\Z\) | Not in database |
$18$ | 18.0.41451359947637504606208.2 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.1999004627104432128.1 | \(\Z/42\Z\) | Not in database |
$18$ | 18.0.2529990231179046912.1 | \(\Z/3\Z \oplus \Z/21\Z\) | Not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 |
---|---|---|---|---|
Reduction type | nonsplit | add | ss | ord |
$\lambda$-invariant(s) | 4 | - | 0,0 | 0 |
$\mu$-invariant(s) | 0 | - | 0,0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.