Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-12716x+550256\)
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(homogenize, simplify) |
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\(y^2z=x^3-12716xz^2+550256z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-12716x+550256\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(85, 289)$ | $1.2564800903790910842492453049$ | $\infty$ |
| $(374, 6936)$ | $1.5093349584874645549547831649$ | $\infty$ |
| $(68, 0)$ | $0$ | $2$ |
Integral points
\( \left(68, 0\right) \), \((70,\pm 56)\), \((85,\pm 289)\), \((136,\pm 1156)\), \((166,\pm 1736)\), \((374,\pm 6936)\), \((646,\pm 16184)\), \((6205,\pm 488699)\), \((216550,\pm 100771384)\)
Invariants
| Conductor: | $N$ | = | \( 18496 \) | = | $2^{6} \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $790939860992$ | = | $2^{15} \cdot 17^{6} $ |
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| j-invariant: | $j$ | = | \( 287496 \) | = | $2^{3} \cdot 3^{3} \cdot 11^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[\sqrt{-4}]\) (potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1457945169565164859983404078$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1372461307715231908979670530$ |
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| $abc$ quality: | $Q$ | ≈ | $1.172456969504371$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.067602236303621$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.7749187956008386780556749807$ |
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| Real period: | $\Omega$ | ≈ | $0.89935832144665119549453741037$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $6.3851519548627282465954019140 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.385151955 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.899358 \cdot 1.774919 \cdot 16}{2^2} \\ & \approx 6.385151955\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 20480 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 6 | 15 | 0 |
| $17$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
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 | $4$ | \( 289 = 17^{2} \) |
| $17$ | additive | $146$ | \( 64 = 2^{6} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 18496i
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 32a3, its twist by $-136$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{34}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1401249857536.17 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.350312464384.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.350312464384.18 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.47883334975488.3 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.2736816128000.1 | \(\Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/20\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ss | ord | ss | ss | ord | add | ss | ss | ord | ss | ord | ord | ss | ss |
| $\lambda$-invariant(s) | - | 2,4 | 2 | 2,2 | 2,2 | 2 | - | 4,2 | 2,2 | 2 | 2,2 | 2 | 2 | 2,2 | 2,2 |
| $\mu$-invariant(s) | - | 0,0 | 0 | 0,0 | 0,0 | 0 | - | 0,0 | 0,0 | 0 | 0,0 | 0 | 0 | 0,0 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.