Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-47916x\)
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(homogenize, simplify) |
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\(y^2z=x^3-47916xz^2\)
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(dehomogenize, simplify) |
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\(y^2=x^3-47916x\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{4761}{16}, \frac{222525}{64}\right) \) | $7.6404279444167974412090515921$ | $\infty$ |
| \( \left(0, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([19044:222525:64]\) | $7.6404279444167974412090515921$ | $\infty$ |
| \([0:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{4761}{16}, \frac{222525}{64}\right) \) | $7.6404279444167974412090515921$ | $\infty$ |
| \( \left(0, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(0, 0\right) \)
\([0:0:1]\)
\( \left(0, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 34848 \) | = | $2^{5} \cdot 3^{2} \cdot 11^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $7040794078162944$ | = | $2^{12} \cdot 3^{6} \cdot 11^{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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7303418535812685449090373401$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.3105329259115095182522750833$ |
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| $abc$ quality: | $Q$ | ≈ | $$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.201765315490791$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.6404279444167974412090515921$ |
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| Real period: | $\Omega$ | ≈ | $0.35444763003463480800006808082$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.4162631546978606802396865787 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.416263155 \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 0.354448 \cdot 7.640428 \cdot 8}{2^2} \\ & \approx 5.416263155\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 168960 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{3}^{*}$ | additive | -1 | 5 | 12 | 0 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 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 | $2$ | \( 99 = 3^{2} \cdot 11 \) |
| $3$ | additive | $6$ | \( 3872 = 2^{5} \cdot 11^{2} \) |
| $11$ | additive | $32$ | \( 288 = 2^{5} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 34848br
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 3872h2, its twist by $33$.
The minimal quartic twist of this elliptic curve is 32.a3, its quartic twist by $396$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{11}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.191664.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.587761422336.37 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.37616731029504.5 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.253912934449152.3 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | 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 |
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 | add | ord | ss | add | ord | ord | ss | ss | ord | ss | ord | ord | ss | ss |
| $\lambda$-invariant(s) | - | - | 1 | 1,1 | - | 1 | 3 | 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 |
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.