Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-194593x+33105057\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-194593xz^2+33105057z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-15762060x+24086300400\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(537, 9120)$ | $2.9289318339967211072809360498$ | $\infty$ |
Integral points
\((537,\pm 9120)\)
Invariants
| Conductor: | $N$ | = | \( 462400 \) | = | $2^{6} \cdot 5^{2} \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-5473632256000$ | = | $-1 \cdot 2^{19} \cdot 5^{3} \cdot 17^{4} $ |
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| j-invariant: | $j$ | = | \( -\frac{297756989}{2} \) | = | $-1 \cdot 2^{-1} \cdot 17^{2} \cdot 101^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6261048251065246482498546943$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.76037987186065710294269486048$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9854089913222234$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6912704019225555$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.9289318339967211072809360498$ |
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| Real period: | $\Omega$ | ≈ | $0.68080995259187708329404592900$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $15.952367544385178370227047402 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 15.952367544 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.680810 \cdot 2.928932 \cdot 8}{1^2} \\ & \approx 15.952367544\end{aligned}$$
Modular invariants
Modular form 462400.2.a.ip
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2350080 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{9}^{*}$ | additive | 1 | 6 | 19 | 1 |
| $5$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $17$ | $1$ | $IV$ | additive | -1 | 2 | 4 | 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 |
|---|---|---|
| $17$ | 17B.4.6 | 17.72.1.4 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 680 = 2^{3} \cdot 5 \cdot 17 \), index $576$, genus $17$, and generators
$\left(\begin{array}{rr} 103 & 34 \\ 119 & 579 \end{array}\right),\left(\begin{array}{rr} 137 & 0 \\ 0 & 273 \end{array}\right),\left(\begin{array}{rr} 1 & 136 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 545 & 136 \\ 544 & 545 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 41 & 272 \\ 136 & 593 \end{array}\right),\left(\begin{array}{rr} 89 & 34 \\ 17 & 115 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 170 & 1 \end{array}\right),\left(\begin{array}{rr} 611 & 374 \\ 595 & 101 \end{array}\right),\left(\begin{array}{rr} 256 & 425 \\ 119 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 170 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 511 & 510 \\ 170 & 171 \end{array}\right)$.
The torsion field $K:=\Q(E[680])$ is a degree-$100270080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/680\Z)$.
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 | $4$ | \( 1445 = 5 \cdot 17^{2} \) |
| $5$ | additive | $10$ | \( 18496 = 2^{6} \cdot 17^{2} \) |
| $17$ | additive | $114$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
17.
Its isogeny class 462400.ip
consists of 2 curves linked by isogenies of
degree 17.
Twists
The minimal quadratic twist of this elliptic curve is 14450.b2, its twist by $8$.
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.