Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-6561x+201439\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-6561xz^2+201439z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-531468x+148443408\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(39, 80\right) \) | $0.80486248315493851671593688274$ | $\infty$ |
| \( \left(49, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([39:80:1]\) | $0.80486248315493851671593688274$ | $\infty$ |
| \([49:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(354, 2160\right) \) | $0.80486248315493851671593688274$ | $\infty$ |
| \( \left(444, 0\right) \) | $0$ | $2$ |
Integral points
\((-25,\pm 592)\), \((39,\pm 80)\), \( \left(49, 0\right) \), \((65,\pm 232)\)
\([-25:\pm 592:1]\), \([39:\pm 80:1]\), \([49:0:1]\), \([65:\pm 232:1]\)
\((-25,\pm 592)\), \((39,\pm 80)\), \( \left(49, 0\right) \), \((65,\pm 232)\)
Invariants
| Conductor: | $N$ | = | \( 438080 \) | = | $2^{6} \cdot 5 \cdot 37^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $165979750400$ | = | $2^{17} \cdot 5^{2} \cdot 37^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{4705274}{25} \) | = | $2 \cdot 5^{-2} \cdot 7^{3} \cdot 19^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.99747054366523254224188749951$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.88721744028907942385788192365$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8418129992365262$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.9237869023287715$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.80486248315493851671593688274$ |
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| Real period: | $\Omega$ | ≈ | $1.0253629926184478165668352626$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.3011048174962512096785120337 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.301104817 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.025363 \cdot 0.804862 \cdot 16}{2^2} \\ & \approx 3.301104817\end{aligned}$$
Modular invariants
Modular form 438080.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 847872 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{7}^{*}$ | additive | -1 | 6 | 17 | 0 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $37$ | $2$ | $III$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1480 = 2^{3} \cdot 5 \cdot 37 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 739 & 0 \end{array}\right),\left(\begin{array}{rr} 297 & 4 \\ 594 & 9 \end{array}\right),\left(\begin{array}{rr} 1044 & 1 \\ 479 & 0 \end{array}\right),\left(\begin{array}{rr} 1297 & 186 \\ 184 & 1295 \end{array}\right),\left(\begin{array}{rr} 1477 & 4 \\ 1476 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[1480])$ is a degree-$111954493440$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1480\Z)$.
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 | $4$ | \( 37 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 87616 = 2^{6} \cdot 37^{2} \) |
| $37$ | additive | $362$ | \( 320 = 2^{6} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 438080h
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 54760b2, its twist by $-8$.
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.