Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-7947x+273782\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-7947xz^2+273782z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-10298691x+12804480702\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-25, 688)$ | $1.0002945245187000147474701207$ | $\infty$ |
Integral points
\( \left(-25, 688\right) \), \( \left(-25, -664\right) \), \( \left(70, 213\right) \), \( \left(70, -284\right) \)
Invariants
| Conductor: | $N$ | = | \( 462722 \) | = | $2 \cdot 13^{2} \cdot 37^{2}$ |
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| Discriminant: | $\Delta$ | = | $-422905697344$ | = | $-1 \cdot 2^{6} \cdot 13^{6} \cdot 37^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{8398297}{64} \) | = | $-1 \cdot 2^{-6} \cdot 37 \cdot 61^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0608006051899779963767631715$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.82349372564816111237799649446$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9122311702877938$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.9565736765846435$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0002945245187000147474701207$ |
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| Real period: | $\Omega$ | ≈ | $0.94842418192979282060348629769$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.5896281288439931844641803175 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.589628129 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.948424 \cdot 1.000295 \cdot 8}{1^2} \\ & \approx 7.589628129\end{aligned}$$
Modular invariants
Modular form 462722.2.a.a
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1327104 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $13$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $37$ | $1$ | $II$ | additive | 1 | 2 | 2 | 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 |
|---|---|---|
| $2$ | 2G | 4.2.0.1 |
| $3$ | 3B | 9.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 17316 = 2^{2} \cdot 3^{2} \cdot 13 \cdot 37 \), index $288$, genus $6$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 36 \\ 12 & 433 \end{array}\right),\left(\begin{array}{rr} 8659 & 9360 \\ 4329 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 15809 & 2691 \\ 17121 & 3836 \end{array}\right),\left(\begin{array}{rr} 13 & 6 \\ 17266 & 17293 \end{array}\right),\left(\begin{array}{rr} 7697 & 9360 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 19 & 27 \\ 15309 & 14464 \end{array}\right),\left(\begin{array}{rr} 17281 & 36 \\ 17280 & 37 \end{array}\right),\left(\begin{array}{rr} 2663 & 0 \\ 0 & 17315 \end{array}\right)$.
The torsion field $K:=\Q(E[17316])$ is a degree-$61891242835968$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/17316\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 231361 = 13^{2} \cdot 37^{2} \) |
| $3$ | good | $2$ | \( 231361 = 13^{2} \cdot 37^{2} \) |
| $13$ | additive | $86$ | \( 2738 = 2 \cdot 37^{2} \) |
| $37$ | additive | $254$ | \( 338 = 2 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 462722.a
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 2738.c1, its twist by $13$.
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.