Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+69x+146\)
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(homogenize, simplify) |
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\(y^2z=x^3+69xz^2+146z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+69x+146\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-2, 0)$ | $0$ | $2$ |
Integral points
\( \left(-2, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 288 \) | = | $2^{5} \cdot 3^{2}$ |
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| Discriminant: | $\Delta$ | = | $-30233088$ | = | $-1 \cdot 2^{9} \cdot 3^{10} $ |
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| j-invariant: | $j$ | = | \( \frac{97336}{81} \) | = | $2^{3} \cdot 3^{-4} \cdot 23^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.12308577383088869731200785667$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.94608075592312513044853885288$ |
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| $abc$ quality: | $Q$ | ≈ | $1.059886267720316$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.2938534767828225$ | |||
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: | $\Omega$ | ≈ | $1.3526394018325432850998294030$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.3526394018325432850998294030 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.352639402 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.352639 \cdot 1.000000 \cdot 4}{2^2} \\ & \approx 1.352639402\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 64 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_0^{*}$ | additive | 1 | 5 | 9 | 0 |
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
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 | 8.24.0.106 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 24.48.0-24.n.1.8, level \( 24 = 2^{3} \cdot 3 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 0 & 17 \\ 19 & 6 \end{array}\right),\left(\begin{array}{rr} 7 & 16 \\ 4 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 18 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 0 & 7 \\ 23 & 12 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 17 & 8 \\ 16 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[24])$ is a degree-$1536$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24\Z)$.
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 | $4$ | \( 9 = 3^{2} \) |
| $3$ | additive | $8$ | \( 32 = 2^{5} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 288c
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 96b4, its twist by $-3$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.0.8.1-2592.3-g3 |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/4\Z\) | 2.2.12.1-192.1-a2 |
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{3})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.2.4608.1 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.339738624.4 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1358954496.8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.339738624.6 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.11609505792.3 | \(\Z/6\Z\) | not in database |
| $16$ | 16.0.1846757322198614016.5 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | 16.0.1846757322198614016.7 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.4.153177439332441840943104.70 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 |
|---|---|---|
| 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 $0$.