Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3-x^2-887x-10143\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3-x^2z-887xz^2-10143z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1149984x-487018224\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(81, 665\right)\) |
$\hat{h}(P)$ | ≈ | $0.98763671776759498658933399182$ |
Integral points
\( \left(35, 21\right) \), \( \left(35, -22\right) \), \( \left(81, 665\right) \), \( \left(81, -666\right) \)
Invariants
Conductor: | \( 121 \) | = | $11^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-2357947691 $ | = | $-1 \cdot 11^{9} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( -32768 \) | = | $-1 \cdot 2^{15}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
|
Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z[(1+\sqrt{-11})/2]\) | (potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
Sato-Tate group: | $N(\mathrm{U}(1))$ | |||
Faltings height: | $0.57587481206971082235980042884\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $-1.2225466425290670856866572546\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $1.0251241218312794\dots$ | |||
Szpiro ratio: | $6.678702041910912\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.98763671776759498658933399182\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.43658375654545366323105953330\dots$ | 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: | $ 2 $ = $ 2 $ | 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: | $1$ | 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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
Special value: | $ L'(E,1) $ ≈ $ 0.86237229669039723995955162179 $ | 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
$\displaystyle 0.862372297 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.436584 \cdot 0.987637 \cdot 2}{1^2} \approx 0.862372297$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 44 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There is only one prime of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$11$ | $2$ | $III^{*}$ | Additive | 1 | 2 | 9 | 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 |
---|---|---|
$11$ | 11B.1.8 | 11.120.1.10 |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
11.
Its isogeny class 121.b
consists of 2 curves linked by isogenies of
degree 11.
Twists
The minimal quadratic twist of this elliptic curve is 121.b2, its twist by $-11$.
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 |
---|---|---|---|
$3$ | 3.1.44.1 | \(\Z/2\Z\) | Not in database |
$4$ | 4.2.11979.1 | \(\Z/3\Z\) | Not in database |
$4$ | 4.0.3993.1 | \(\Z/3\Z\) | Not in database |
$6$ | 6.0.21296.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$8$ | 8.0.143496441.1 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$8$ | 8.0.221445125.1 | \(\Z/5\Z\) | Not in database |
$10$ | \(\Q(\zeta_{11})\) | \(\Z/11\Z\) | Not in database |
$12$ | 12.2.2472487358038016.1 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/9\Z\) | Not in database |
$12$ | 12.2.440049629885184.1 | \(\Z/6\Z\) | Not in database |
$12$ | 12.0.16298134440192.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | 16.4.766217865410400390625.1 | \(\Z/5\Z\) | Not in database |
$16$ | deg 16 | \(\Z/15\Z\) | Not in database |
$20$ | 20.0.14861658978964734748713.1 | \(\Z/33\Z\) | 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.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | ss | ord | ord | ss | add | ss | ss | ss | ord | ss | ord | ord | ss | ss | ord |
$\lambda$-invariant(s) | ? | 1 | 1 | 1,1 | - | 1,1 | 1,1 | 1,1 | 1 | 1,1 | 1 | 1 | 1,1 | 1,1 | 1 |
$\mu$-invariant(s) | ? | 0 | 0 | 0,0 | - | 0,0 | 0,0 | 0,0 | 0 | 0,0 | 0 | 0 | 0,0 | 0,0 | 0 |
An entry ? indicates that the invariants have not yet been computed.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.