This is the modular curve for the level subgroup of level 16, denoted by 16B^1-16c in [MR:3671434]. It is also isomorphic to the modular curves $X(16F^1$-$16a)$ and $X(16F^1$-$16h)$.
Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-2x\)
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(homogenize, simplify) |
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\(y^2z=x^3-2xz^2\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2x\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2, 2)$ | $0.60870903197698136089719041959$ | $\infty$ |
| $(0, 0)$ | $0$ | $2$ |
Integral points
\((-1,\pm 1)\), \( \left(0, 0\right) \), \((2,\pm 2)\), \((338,\pm 6214)\)
Invariants
| Conductor: | $N$ | = | \( 256 \) | = | $2^{8}$ |
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| Discriminant: | $\Delta$ | = | $512$ | = | $2^{9} $ |
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| j-invariant: | $j$ | = | \( 1728 \) | = | $2^{6} \cdot 3^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[\sqrt{-1}]\) (potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.79067254049155053618935099221$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.3105329259115095182522750833$ |
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| $abc$ quality: | $Q$ | ≈ | $$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.4693609377704338$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.60870903197698136089719041959$ |
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| Real period: | $\Omega$ | ≈ | $4.4097575959863310911177975020$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.3421296387529900325364678915 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.342129639 \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 4.409758 \cdot 0.608709 \cdot 2}{2^2} \\ & \approx 1.342129639\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 8 |
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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 is only one prime $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | 1 | 8 | 9 | 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 |
|---|---|---|
| $2$ | 2B | 16.384.9.831 |
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 | $4$ | \( 1 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 256b
consists of 2 curves linked by isogenies of
degree 2.
Twists
This elliptic curve is its own minimal quadratic twist.
The minimal quartic twist of this elliptic curve is 32.a3, its quartic twist by $8$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.2.8.1-1024.1-k4 |
| $4$ | 4.0.2048.1 | \(\Z/4\Z\) | not in database |
| $4$ | 4.2.2048.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.16777216.2 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.67108864.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.36691771392.3 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.2097152000.8 | \(\Z/10\Z\) | not in database |
| $16$ | 16.0.18014398509481984.1 | \(\Z/8\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.0.1346286087882789617664.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/10\Z\) | not in database |
| $16$ | 16.4.5385144351531158470656.13 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/10\Z\) | 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 | ord | ss | ss | ord | ss | ord | ord | ss | ss |
| $\lambda$-invariant(s) | - | 1,7 | 5 | 1,1 | 1,1 | 3 | 1 | 1,1 | 1,1 | 1 | 1,1 | 1 | 1 | 1,1 | 1,1 |
| $\mu$-invariant(s) | - | 0,0 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.