Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-115233x-8616609\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-115233xz^2-8616609z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-9333900x-6253506288\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(999, 29568)$ | $2.3673189499426244300210460369$ | $\infty$ |
| $(-79, 0)$ | $0$ | $2$ |
Integral points
\( \left(-79, 0\right) \), \((999,\pm 29568)\)
Invariants
| Conductor: | $N$ | = | \( 27456 \) | = | $2^{6} \cdot 3 \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $66140755734822912$ | = | $2^{22} \cdot 3^{3} \cdot 11^{2} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{645532578015625}{252306960048} \) | = | $2^{-4} \cdot 3^{-3} \cdot 5^{6} \cdot 11^{-2} \cdot 13^{-6} \cdot 3457^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9262393452481473066187550419$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.88651857440822934249290685971$ |
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| $abc$ quality: | $Q$ | ≈ | $1.033360901289125$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.557357782032954$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.3673189499426244300210460369$ |
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| Real period: | $\Omega$ | ≈ | $0.26791185713416871417322474477$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2^{2}\cdot3\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.6107937956964643286265860873 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.610793796 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.267912 \cdot 2.367319 \cdot 48}{2^2} \\ & \approx 7.610793796\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 221184 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{12}^{*}$ | additive | -1 | 6 | 22 | 4 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
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 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 937 & 12 \\ 474 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1727 & 3420 \\ 1728 & 3419 \end{array}\right),\left(\begin{array}{rr} 2854 & 3421 \\ 2871 & 1736 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 3382 & 3423 \end{array}\right),\left(\begin{array}{rr} 3421 & 3430 \\ 1766 & 1725 \end{array}\right),\left(\begin{array}{rr} 1715 & 0 \\ 0 & 3431 \end{array}\right),\left(\begin{array}{rr} 2507 & 3420 \\ 3030 & 3359 \end{array}\right),\left(\begin{array}{rr} 1002 & 1585 \\ 1001 & 1574 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 3421 & 12 \\ 3420 & 13 \end{array}\right)$.
The torsion field $K:=\Q(E[3432])$ is a degree-$265686220800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3432\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | additive | $2$ | \( 3 \) |
| $3$ | split multiplicative | $4$ | \( 704 = 2^{6} \cdot 11 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 2496 = 2^{6} \cdot 3 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 2112 = 2^{6} \cdot 3 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 27456.by
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 858.c3, its twist by $-8$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.3926208.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{3})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.809588736.9 | \(\Z/6\Z\) | not in database |
| $8$ | 8.4.3907864756224.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.2219775733334016.24 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.61660437037056.18 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.2208153276245515844253255544530951494877315072.1 | \(\Z/18\Z\) | 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 | add | split | ss | ord | nonsplit | nonsplit | ss | ord | ss | ss | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 2 | 1,1 | 5 | 1 | 5 | 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 | 0 | 0,0 |
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.