Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-222700x+59534000\)
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(homogenize, simplify) |
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\(y^2z=x^3-222700xz^2+59534000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-222700x+59534000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1309/9, 148427/27)$ | $7.0445921120171002418158567547$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 462400 \) | = | $2^{6} \cdot 5^{2} \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-824264622080000000$ | = | $-1 \cdot 2^{31} \cdot 5^{7} \cdot 17^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{60698457}{40960} \) | = | $-1 \cdot 2^{-13} \cdot 3^{3} \cdot 5^{-1} \cdot 131^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1383682333939929346174309245$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.41437482967702923687118057877$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9780347149700016$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.7815516687539112$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.0445921120171002418158567547$ |
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| Real period: | $\Omega$ | ≈ | $0.26037232709940869471892116266$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $14.673734733376246083493482569 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.673734733 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.260372 \cdot 7.044592 \cdot 8}{1^2} \\ & \approx 14.673734733\end{aligned}$$
Modular invariants
Modular form 462400.2.a.iy
For more coefficients, see the Downloads section to the right.
| Modular degree: | 11501568 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{21}^{*}$ | additive | 1 | 6 | 31 | 13 |
| $5$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $17$ | $2$ | $III$ | 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 |
|---|---|---|
| $13$ | 13B.5.1 | 13.42.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8840 = 2^{3} \cdot 5 \cdot 13 \cdot 17 \), index $336$, genus $9$, and generators
$\left(\begin{array}{rr} 3658 & 13 \\ 3471 & 8718 \end{array}\right),\left(\begin{array}{rr} 8815 & 26 \\ 8814 & 27 \end{array}\right),\left(\begin{array}{rr} 8826 & 8827 \\ 4433 & 12 \end{array}\right),\left(\begin{array}{rr} 6631 & 4446 \\ 0 & 4591 \end{array}\right),\left(\begin{array}{rr} 1 & 26 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 14 & 23 \\ 871 & 1431 \end{array}\right),\left(\begin{array}{rr} 3522 & 8827 \\ 1781 & 12 \end{array}\right),\left(\begin{array}{rr} 4434 & 13 \\ 2197 & 8828 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 26 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[8840])$ is a degree-$4504934154240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8840\Z)$.
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 | $4$ | \( 425 = 5^{2} \cdot 17 \) |
| $5$ | additive | $18$ | \( 18496 = 2^{6} \cdot 17^{2} \) |
| $17$ | additive | $82$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
13.
Its isogeny class 462400iy
consists of 2 curves linked by isogenies of
degree 13.
Twists
The minimal quadratic twist of this elliptic curve is 2890j1, its twist by $40$.
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.