Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-6845307771x-217990506787670\)
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(homogenize, simplify) |
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\(y^2z=x^3-6845307771xz^2-217990506787670z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6845307771x-217990506787670\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-47770, 0)$ | $0$ | $2$ |
Integral points
\( \left(-47770, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 457776 \) | = | $2^{4} \cdot 3^{2} \cdot 11 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $122631708372043409915904$ | = | $2^{31} \cdot 3^{8} \cdot 11^{6} \cdot 17^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{2418067440128989194388361}{8359273562112} \) | = | $2^{-19} \cdot 3^{-2} \cdot 11^{-6} \cdot 23^{3} \cdot 1933^{3} \cdot 3019^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.0740647453013725868445581940$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.1233080843933184116673197996$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0715970690308463$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.103518892166107$ | |||
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: | $\Omega$ | ≈ | $0.016597829918899098639412346786$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 192 $ = $ 2^{2}\cdot2^{2}\cdot( 2 \cdot 3 )\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.79669583610715673469179264573 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.796695836 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.016598 \cdot 1.000000 \cdot 192}{2^2} \\ & \approx 0.796695836\end{aligned}$$
Modular invariants
Modular form 457776.2.a.bw
For more coefficients, see the Downloads section to the right.
| Modular degree: | 313786368 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{23}^{*}$ | additive | -1 | 4 | 31 | 19 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $11$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $17$ | $2$ | $III$ | additive | 1 | 2 | 3 | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4488 = 2^{3} \cdot 3 \cdot 11 \cdot 17 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 409 & 4 \\ 818 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 3929 & 562 \\ 560 & 3927 \end{array}\right),\left(\begin{array}{rr} 4485 & 4 \\ 4484 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2993 & 4 \\ 1498 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 4228 & 1 \\ 3431 & 0 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 2243 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[4488])$ is a degree-$6353112268800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4488\Z)$.
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 | $4$ | \( 153 = 3^{2} \cdot 17 \) |
| $3$ | additive | $8$ | \( 4624 = 2^{4} \cdot 17^{2} \) |
| $11$ | split multiplicative | $12$ | \( 41616 = 2^{4} \cdot 3^{2} \cdot 17^{2} \) |
| $17$ | additive | $82$ | \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 457776.bw
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 19074.b1, its twist by $12$.
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 $0$.