Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-1100x-14000\)
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(homogenize, simplify) |
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\(y^2z=x^3-1100xz^2-14000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1100x-14000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-20, 0)$ | $0$ | $2$ |
Integral points
\( \left(-20, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 1600 \) | = | $2^{6} \cdot 5^{2}$ |
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| Discriminant: | $\Delta$ | = | $512000000$ | = | $2^{15} \cdot 5^{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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.53390680114545863317395276550$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1372461307715231908979670529$ |
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| $abc$ quality: | $Q$ | ≈ | $1.172456969504371$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.421776289369018$ | |||
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: | $\Omega$ | ≈ | $0.82916740276137078901307561270$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.6583348055227415780261512254 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.658334806 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.829167 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 1.658334806\end{aligned}$$
Modular invariants
\( q - 3 q^{9} + 6 q^{13} - 2 q^{17} + O(q^{20}) \)
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For more coefficients, see the Downloads section to the right.
| Modular degree: | 512 |
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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 |
| $5$ | $2$ | $I_0^{*}$ | 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 | $4$ | \( 25 = 5^{2} \) |
| $5$ | additive | $14$ | \( 64 = 2^{6} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 1600o
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 $40$.
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 |
| $2$ | \(\Q(\sqrt{-10}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-5})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.2621440000.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.2621440000.8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.10485760000.2 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.358318080000.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.819200000.1 | \(\Z/10\Z\) | not in database |
| $16$ | 16.0.109951162777600000000.3 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | 16.0.6871947673600000000.6 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $16$ | 16.4.16777216000000000000.2 | \(\Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | 16.0.10737418240000000000.2 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | 16.0.10737418240000000000.1 | \(\Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | 16.0.671088640000000000.2 | \(\Z/20\Z\) | 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 | 5 |
|---|---|---|
| Reduction type | add | add |
| $\lambda$-invariant(s) | - | - |
| $\mu$-invariant(s) | - | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.