Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-690592884195x-220900315730478878\)
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(homogenize, simplify) |
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\(y^2z=x^3-690592884195xz^2-220900315730478878z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-690592884195x-220900315730478878\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(62272790796454693532785975360999487/795115419633012243019309441, 15538995645101295461187639587337610346376749778834930/22420498456868347274965664470597594663489)$ | $74.948661118490447681577817004$ | $\infty$ |
| $(959582, 0)$ | $0$ | $2$ |
Integral points
\( \left(959582, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 457776 \) | = | $2^{4} \cdot 3^{2} \cdot 11 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-1463342274693297285835943566245888$ | = | $-1 \cdot 2^{17} \cdot 3^{8} \cdot 11^{2} \cdot 17^{18} $ |
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| j-invariant: | $j$ | = | \( -\frac{505369473241574671219626625}{20303219722982711328} \) | = | $-1 \cdot 2^{-5} \cdot 3^{-2} \cdot 5^{3} \cdot 11^{-2} \cdot 17^{-12} \cdot 19^{3} \cdot 8384543^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $5.4465784718611551467580672196$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.7875184749390469515184451707$ |
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| $abc$ quality: | $Q$ | ≈ | $1.103954672669075$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.1655019332194865$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $74.948661118490447681577817004$ |
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| Real period: | $\Omega$ | ≈ | $0.0026185627608978659468055376179$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2^{2}\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.2802487354890522973334022237 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.280248735 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.002619 \cdot 74.948661 \cdot 128}{2^2} \\ & \approx 6.280248735\end{aligned}$$
Modular invariants
Modular form 457776.2.a.dd
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3822059520 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{9}^{*}$ | additive | -1 | 4 | 17 | 5 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $17$ | $4$ | $I_{12}^{*}$ | additive | 1 | 2 | 18 | 12 |
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 | 8.6.0.5 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4488 = 2^{3} \cdot 3 \cdot 11 \cdot 17 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 4438 & 4479 \end{array}\right),\left(\begin{array}{rr} 2430 & 527 \\ 187 & 934 \end{array}\right),\left(\begin{array}{rr} 3431 & 0 \\ 0 & 4487 \end{array}\right),\left(\begin{array}{rr} 1495 & 1836 \\ 2414 & 2039 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 409 & 2652 \\ 3774 & 2449 \end{array}\right),\left(\begin{array}{rr} 4477 & 12 \\ 4476 & 13 \end{array}\right),\left(\begin{array}{rr} 4234 & 2907 \\ 3009 & 256 \end{array}\right)$.
The torsion field $K:=\Q(E[4488])$ is a degree-$794139033600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4488\Z)$.
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 | $4$ | \( 2601 = 3^{2} \cdot 17^{2} \) |
| $3$ | additive | $8$ | \( 50864 = 2^{4} \cdot 11 \cdot 17^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 41616 = 2^{4} \cdot 3^{2} \cdot 17^{2} \) |
| $17$ | additive | $162$ | \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 457776.dd
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 1122.j2, its twist by $204$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.