Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-432630x-107155049\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-432630xz^2-107155049z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6922083x-6864845218\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-334, 167)$ | $0$ | $2$ |
| $(758, -379)$ | $0$ | $2$ |
Integral points
\( \left(-334, 167\right) \), \( \left(758, -379\right) \)
Invariants
| Conductor: | $N$ | = | \( 487305 \) | = | $3^{2} \cdot 5 \cdot 7^{2} \cdot 13 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $212063220235400625$ | = | $3^{10} \cdot 5^{4} \cdot 7^{6} \cdot 13^{2} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{104413920565969}{2472575625} \) | = | $3^{-4} \cdot 5^{-4} \cdot 7^{6} \cdot 13^{-2} \cdot 17^{-2} \cdot 31^{6}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1098490664584600306705173296$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.58758784759674853242021833942$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1569603776083688$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.859501182077428$ | |||
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: | $\Omega$ | ≈ | $0.18642187534863002571400614783$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.4913750027890402057120491826 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.491375003 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.186422 \cdot 1.000000 \cdot 128}{4^2} \\ & \approx 1.491375003\end{aligned}$$
Modular invariants
Modular form 487305.2.a.cx
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4718592 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $17$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 18564 = 2^{2} \cdot 3 \cdot 7 \cdot 13 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 8569 & 17682 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9283 & 17682 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 18561 & 4 \\ 18560 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5303 & 0 \\ 0 & 18563 \end{array}\right),\left(\begin{array}{rr} 10039 & 17682 \\ 18354 & 883 \end{array}\right),\left(\begin{array}{rr} 6187 & 0 \\ 0 & 18563 \end{array}\right)$.
The torsion field $K:=\Q(E[18564])$ is a degree-$397335192403968$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/18564\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 441 = 3^{2} \cdot 7^{2} \) |
| $3$ | additive | $8$ | \( 54145 = 5 \cdot 7^{2} \cdot 13 \cdot 17 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 97461 = 3^{2} \cdot 7^{2} \cdot 13 \cdot 17 \) |
| $7$ | additive | $26$ | \( 9945 = 3^{2} \cdot 5 \cdot 13 \cdot 17 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 37485 = 3^{2} \cdot 5 \cdot 7^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 28665 = 3^{2} \cdot 5 \cdot 7^{2} \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 487305.cx
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 3315.b2, its twist by $21$.
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 $0$.