Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-7203x+1050658\)
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(homogenize, simplify) |
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\(y^2z=x^3-7203xz^2+1050658z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7203x+1050658\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-49, 1134)$ | $0.69555816332121850521330991852$ | $\infty$ |
| $(113, 1296)$ | $1.0297301135798382348560040830$ | $\infty$ |
Integral points
\((-82,\pm 1044)\), \((-63,\pm 1120)\), \((-49,\pm 1134)\), \((98,\pm 1134)\), \((113,\pm 1296)\), \((161,\pm 2016)\), \((2241,\pm 106016)\), \((4487,\pm 300510)\), \((26033,\pm 4200336)\)
Invariants
| Conductor: | $N$ | = | \( 458640 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-452959380357120$ | = | $-1 \cdot 2^{14} \cdot 3^{11} \cdot 5 \cdot 7^{4} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{5764801}{63180} \) | = | $-1 \cdot 2^{-2} \cdot 3^{-5} \cdot 5^{-1} \cdot 7^{8} \cdot 13^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4942732774212259877122599375$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.39681676382454526910437905023$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0795661647694836$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.16453926713549$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.67051537696014746064783460349$ |
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| Real period: | $\Omega$ | ≈ | $0.44901724301609193429368406066$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2^{2}\cdot2^{2}\cdot1\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $14.451502366201969106039480630 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.451502366 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.449017 \cdot 0.670515 \cdot 48}{1^2} \\ & \approx 14.451502366\end{aligned}$$
Modular invariants
Modular form 458640.2.a.m
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1751040 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{6}^{*}$ | additive | -1 | 4 | 14 | 2 |
| $3$ | $4$ | $I_{5}^{*}$ | additive | -1 | 2 | 11 | 5 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $3$ | $IV$ | additive | 1 | 2 | 4 | 0 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 301 & 2 \\ 301 & 3 \end{array}\right),\left(\begin{array}{rr} 389 & 2 \\ 388 & 3 \end{array}\right),\left(\begin{array}{rr} 131 & 2 \\ 131 & 3 \end{array}\right),\left(\begin{array}{rr} 157 & 2 \\ 157 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 389 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[390])$ is a degree-$1811496960$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/390\Z)$.
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 | $2$ | \( 28665 = 3^{2} \cdot 5 \cdot 7^{2} \cdot 13 \) |
| $3$ | additive | $8$ | \( 50960 = 2^{4} \cdot 5 \cdot 7^{2} \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 91728 = 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $20$ | \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 35280 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 458640.m consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 19110.bx1, its twist by $12$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.