Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-317898x-69307441\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-317898xz^2-69307441z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5086368x-4435676208\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(22011/25, 2281991/125)$ | $5.2989597277002009645390764122$ | $\infty$ |
| $(7453/9, 414179/27)$ | $5.8808802011851039629373694573$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 491985 \) | = | $3^{2} \cdot 5 \cdot 13 \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $-19025349569269875$ | = | $-1 \cdot 3^{9} \cdot 5^{3} \cdot 13 \cdot 29^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{303464448}{1625} \) | = | $-1 \cdot 2^{15} \cdot 3^{3} \cdot 5^{-3} \cdot 7^{3} \cdot 13^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9682394458687130583206613485$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.53936768562560622381740859537$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9400430053541569$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.7868533910202946$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $30.611644769552298034121690562$ |
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| Real period: | $\Omega$ | ≈ | $0.10049826087279219899438723679$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $12.305668247182846738732932252 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 12.305668247 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.100498 \cdot 30.611645 \cdot 4}{1^2} \\ & \approx 12.305668247\end{aligned}$$
Modular invariants
Modular form 491985.2.a.t
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3483648 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $29$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 11310 = 2 \cdot 3 \cdot 5 \cdot 13 \cdot 29 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 10441 & 1566 \\ 9483 & 4699 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 11305 & 6 \\ 11304 & 7 \end{array}\right),\left(\begin{array}{rr} 7799 & 0 \\ 0 & 11309 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 6787 & 1566 \\ 9831 & 4699 \end{array}\right),\left(\begin{array}{rr} 1884 & 7859 \\ 2987 & 8496 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[11310])$ is a degree-$154448230809600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/11310\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 163995 = 3 \cdot 5 \cdot 13 \cdot 29^{2} \) |
| $3$ | additive | $2$ | \( 10933 = 13 \cdot 29^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 98397 = 3^{2} \cdot 13 \cdot 29^{2} \) |
| $13$ | split multiplicative | $14$ | \( 37845 = 3^{2} \cdot 5 \cdot 29^{2} \) |
| $29$ | additive | $422$ | \( 585 = 3^{2} \cdot 5 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 491985.t
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 585.e1, its twist by $29$.
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.