Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-6838896x-8062449172\)
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(homogenize, simplify) |
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\(y^2z=x^3-6838896xz^2-8062449172z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6838896x-8062449172\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 457776 \) | = | $2^{4} \cdot 3^{2} \cdot 11 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-7610384626768305955584$ | = | $-1 \cdot 2^{8} \cdot 3^{18} \cdot 11 \cdot 17^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{27172077568}{5845851} \) | = | $-1 \cdot 2^{16} \cdot 3^{-12} \cdot 11^{-1} \cdot 17 \cdot 29^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9197714282922624975702466111$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.019558267547433392094779499752$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9854098358293056$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.537632208444637$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.046148137823695340274688830507$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2^{2}\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $5.9069616414330035551601703049 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $16$ = $4^2$ (exact) |
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BSD formula
$$\begin{aligned} 5.906961641 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{16 \cdot 0.046148 \cdot 1.000000 \cdot 8}{1^2} \\ & \approx 5.906961641\end{aligned}$$
Modular invariants
Modular form 457776.2.a.fr
For more coefficients, see the Downloads section to the right.
| Modular degree: | 33841152 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | -1 | 4 | 8 | 0 |
| $3$ | $4$ | $I_{12}^{*}$ | additive | -1 | 2 | 18 | 12 |
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $17$ | $1$ | $IV^{*}$ | additive | -1 | 2 | 8 | 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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 132 = 2^{2} \cdot 3 \cdot 11 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 13 & 6 \\ 105 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 120 & 71 \\ 77 & 21 \end{array}\right),\left(\begin{array}{rr} 71 & 126 \\ 72 & 125 \end{array}\right),\left(\begin{array}{rr} 129 & 130 \\ 56 & 125 \end{array}\right),\left(\begin{array}{rr} 65 & 0 \\ 0 & 131 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 127 & 6 \\ 126 & 7 \end{array}\right)$.
The torsion field $K:=\Q(E[132])$ is a degree-$3801600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/132\Z)$.
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 | $2$ | \( 28611 = 3^{2} \cdot 11 \cdot 17^{2} \) |
| $3$ | additive | $2$ | \( 50864 = 2^{4} \cdot 11 \cdot 17^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 41616 = 2^{4} \cdot 3^{2} \cdot 17^{2} \) |
| $17$ | additive | $114$ | \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 457776fr
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 38148f1, its twist by $204$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.