Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-1124065x+458394337\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-1124065xz^2+458394337z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-91049292x+333896323824\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(424, 7605\right)\)
|
| $\hat{h}(P)$ | ≈ | $0.49382041103376956989013427096$ |
Torsion generators
\( \left(593, 0\right) \)
Integral points
\((-1071,\pm 20800)\), \((64,\pm 19665)\), \((424,\pm 7605)\), \( \left(593, 0\right) \), \((1009,\pm 18720)\)
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 212160 \) | = | $2^{6} \cdot 3 \cdot 5 \cdot 13 \cdot 17$ |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | $272241295073280000 $ | = | $2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 13^{6} \cdot 17 $ |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
| j-invariant: | \( \frac{2396726313900986596}{4154072495625} \) | = | $2^{2} \cdot 3^{-4} \cdot 5^{-4} \cdot 13^{-6} \cdot 17^{-1} \cdot 19^{3} \cdot 44371^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $2.2393375922581837715701781561\dots$ | ||
| Stable Faltings height: | $1.3151413515115900256805353275\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
| |||
| Analytic rank: | $1$ | ||
sage: E.regulator()
magma: Regulator(E);
| |||
| Regulator: | $0.49382041103376956989013427096\dots$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
| |||
| Real period: | $0.30956310408703058585904038157\dots$ | ||
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: | $ 192 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot( 2 \cdot 3 )\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) $ ≈ $ 7.3376918064550577547530879346 $ | ||
Modular invariants
Modular form 212160.2.a.cy
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 2949120 | ||
| $ \Gamma_0(N) $-optimal: | yes | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{6}^{*}$ | Additive | 1 | 6 | 16 | 0 |
| $3$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
| $5$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
| $13$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
| $17$ | $1$ | $I_{1}$ | 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 | 2.3.0.1 |
$p$-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 non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 212160.cy
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
| $4$ | 4.0.735488.6 | \(\Z/4\Z\) | Not in database |
| $8$ | 8.4.16708604563456.43 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | 8.0.156332410863616.15 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/6\Z\) | 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.