Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-121060x-16170740\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-121060xz^2-16170740z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-9805887x-11817887094\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(369151917/763876, 4113031826617/667627624)$ | $20.073835086134215002903806086$ | $\infty$ |
| $(-199, 0)$ | $0$ | $2$ |
Integral points
\( \left(-199, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 446160 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 11 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $20184556803840$ | = | $2^{8} \cdot 3^{3} \cdot 5 \cdot 11^{2} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{158792223184}{16335} \) | = | $2^{4} \cdot 3^{-3} \cdot 5^{-1} \cdot 7^{3} \cdot 11^{-2} \cdot 307^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5857763312401009302535829397$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.15879646786396431071798219539$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9214349789031259$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.591957334205622$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $20.073835086134215002903806086$ |
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| Real period: | $\Omega$ | ≈ | $0.25594801696213054839974271446$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot1\cdot1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.1378582831208913996189722157 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.137858283 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.255948 \cdot 20.073835 \cdot 4}{2^2} \\ & \approx 5.137858283\end{aligned}$$
Modular invariants
Modular form 446160.2.a.ci
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2488320 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 4 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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_0^{*}$ | additive | -1 | 4 | 8 | 0 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8580 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 13 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 11 & 2 \\ 8530 & 8571 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 8569 & 12 \\ 8568 & 13 \end{array}\right),\left(\begin{array}{rr} 714 & 923 \\ 715 & 3574 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 7801 & 2652 \\ 5226 & 7333 \end{array}\right),\left(\begin{array}{rr} 1990 & 663 \\ 7761 & 6592 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5939 & 0 \\ 0 & 8579 \end{array}\right),\left(\begin{array}{rr} 1106 & 5291 \\ 429 & 3940 \end{array}\right)$.
The torsion field $K:=\Q(E[8580])$ is a degree-$7970586624000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8580\Z)$.
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 | $2$ | \( 2535 = 3 \cdot 5 \cdot 13^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 148720 = 2^{4} \cdot 5 \cdot 11 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 89232 = 2^{4} \cdot 3 \cdot 11 \cdot 13^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 40560 = 2^{4} \cdot 3 \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 446160.ci
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 660.c2, its twist by $-52$.
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.