Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-3x-2\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-3xz^2-2z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-270x-675\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(3, 5)$ | $0.29321102660236279365450119921$ | $\infty$ |
| $(-2, 0)$ | $0$ | $2$ |
Integral points
\( \left(-2, 0\right) \), \((-1,\pm 1)\), \((2,\pm 2)\), \((3,\pm 5)\), \((43,\pm 285)\)
Invariants
| Conductor: | $N$ | = | \( 200 \) | = | $2^{3} \cdot 5^{2}$ |
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| Discriminant: | $\Delta$ | = | $2000$ | = | $2^{4} \cdot 5^{3} $ |
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| j-invariant: | $j$ | = | \( 2048 \) | = | $2^{11}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.67321922167231579705643901007$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.3066277599674893271790395505$ |
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| $abc$ quality: | $Q$ | ≈ | $1.018975235452531$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.873652216802034$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.29321102660236279365450119921$ |
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| Real period: | $\Omega$ | ≈ | $3.7121152731513333269060680258$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.0884331301070128243531005638 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.088433130 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 3.712115 \cdot 0.293211 \cdot 4}{2^2} \\ & \approx 1.088433130\end{aligned}$$
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 are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | -1 | 3 | 4 | 0 |
| $5$ | $2$ | $III$ | additive | -1 | 2 | 3 | 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 | 16.96.3.338 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 80 = 2^{4} \cdot 5 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 17 & 0 \\ 0 & 33 \end{array}\right),\left(\begin{array}{rr} 49 & 8 \\ 28 & 5 \end{array}\right),\left(\begin{array}{rr} 51 & 42 \\ 74 & 39 \end{array}\right),\left(\begin{array}{rr} 65 & 64 \\ 16 & 65 \end{array}\right),\left(\begin{array}{rr} 31 & 62 \\ 48 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 41 & 40 \\ 40 & 41 \end{array}\right),\left(\begin{array}{rr} 37 & 44 \\ 28 & 29 \end{array}\right),\left(\begin{array}{rr} 22 & 29 \\ 55 & 26 \end{array}\right)$.
The torsion field $K:=\Q(E[80])$ is a degree-$30720$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/80\Z)$.
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 | $2$ | \( 5 \) |
| $5$ | additive | $10$ | \( 8 = 2^{3} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 200.b
consists of 2 curves linked by isogenies of
degree 2.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.2.5.1-1600.1-f3 |
| $4$ | \(\Q(\zeta_{5})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.64000000.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.4000000.2 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.34992000000.2 | \(\Z/6\Z\) | not in database |
| $16$ | 16.0.4096000000000000.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | ord | add | ord | ord | ord | ss | ord | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | - | 1 | 1 | 1 | 1,3 | 1 | 1 | 1 | 3,1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 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.