Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-9939960x-12064646892\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-9939960xz^2-12064646892z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-805136787x-8792712173934\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 446160 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 11 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $8614116413760000000$ | = | $2^{12} \cdot 3 \cdot 5^{7} \cdot 11 \cdot 13^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{32506551525721}{2578125} \) | = | $3^{-1} \cdot 5^{-7} \cdot 7^{6} \cdot 11^{-1} \cdot 13 \cdot 277^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6797665829452030167141131284$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.27665316407756654992788937923$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9880067271115119$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.608535417496817$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.085026726384053739102518662109$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 84 $ = $ 2^{2}\cdot1\cdot7\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $7.1422450162605140846115676172 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 7.142245016 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.085027 \cdot 1.000000 \cdot 84}{1^2} \\ & \approx 7.142245016\end{aligned}$$
Modular invariants
Modular form 446160.2.a.jo
For more coefficients, see the Downloads section to the right.
| Modular degree: | 23482368 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | -1 | 4 | 12 | 0 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 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 |
|---|---|---|
| $7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 60060 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 30043 & 60046 \\ 30044 & 60045 \end{array}\right),\left(\begin{array}{rr} 30029 & 0 \\ 0 & 60059 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 39267 & 60052 \\ 36974 & 30067 \end{array}\right),\left(\begin{array}{rr} 16381 & 14 \\ 24577 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 54053 & 60046 \\ 48041 & 59961 \end{array}\right),\left(\begin{array}{rr} 20021 & 14 \\ 20027 & 99 \end{array}\right),\left(\begin{array}{rr} 60047 & 14 \\ 60046 & 15 \end{array}\right),\left(\begin{array}{rr} 60059 & 60046 \\ 0 & 38609 \end{array}\right)$.
The torsion field $K:=\Q(E[60060])$ is a degree-$16068702633984000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/60060\Z)$.
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 | $2$ | \( 27885 = 3 \cdot 5 \cdot 11 \cdot 13^{2} \) |
| $3$ | split multiplicative | $4$ | \( 148720 = 2^{4} \cdot 5 \cdot 11 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 89232 = 2^{4} \cdot 3 \cdot 11 \cdot 13^{2} \) |
| $7$ | good | $2$ | \( 89232 = 2^{4} \cdot 3 \cdot 11 \cdot 13^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 40560 = 2^{4} \cdot 3 \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $74$ | \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 446160jo
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 27885f1, its twist by $-52$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.