Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-3757176344x-95815305679040\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-3757176344xz^2-95815305679040z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-60114821507x-6132239678280066\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 488410 \) | = | $2 \cdot 5 \cdot 13^{2} \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-571671977025154973980206039040$ | = | $-1 \cdot 2^{35} \cdot 5 \cdot 13^{10} \cdot 17^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{1762712152495281}{171798691840} \) | = | $-1 \cdot 2^{-35} \cdot 3^{3} \cdot 5^{-1} \cdot 13^{2} \cdot 7283^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.4508669024430362100066032216$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.89680243253031422317059637802$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1377369369986685$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.9478269034491165$ | |||
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: | $\Omega$ | ≈ | $0.0095886637875125589293793177677$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.93968905117623077507917314124 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $49$ = $7^2$ (exact) |
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BSD formula
$$\begin{aligned} 0.939689051 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{49 \cdot 0.009589 \cdot 1.000000 \cdot 2}{1^2} \\ & \approx 0.939689051\end{aligned}$$
Modular invariants
Modular form 488410.2.a.u
For more coefficients, see the Downloads section to the right.
| Modular degree: | 753392640 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{35}$ | nonsplit multiplicative | 1 | 1 | 35 | 35 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $1$ | $II^{*}$ | additive | 1 | 2 | 10 | 0 |
| $17$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 61880 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \cdot 17 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 30941 & 36414 \\ 49147 & 7379 \end{array}\right),\left(\begin{array}{rr} 12377 & 36414 \\ 42959 & 7379 \end{array}\right),\left(\begin{array}{rr} 61867 & 14 \\ 61866 & 15 \end{array}\right),\left(\begin{array}{rr} 22949 & 3638 \\ 27370 & 21929 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15471 & 36414 \\ 2737 & 7379 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 58239 & 0 \\ 0 & 61879 \end{array}\right),\left(\begin{array}{rr} 11833 & 29648 \\ 56406 & 58073 \end{array}\right)$.
The torsion field $K:=\Q(E[61880])$ is a degree-$31786815392317440$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/61880\Z)$.
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 | $4$ | \( 244205 = 5 \cdot 13^{2} \cdot 17^{2} \) |
| $5$ | split multiplicative | $6$ | \( 48841 = 13^{2} \cdot 17^{2} \) |
| $7$ | good | $2$ | \( 244205 = 5 \cdot 13^{2} \cdot 17^{2} \) |
| $13$ | additive | $50$ | \( 2890 = 2 \cdot 5 \cdot 17^{2} \) |
| $17$ | additive | $146$ | \( 1690 = 2 \cdot 5 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 488410.u
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 1690.h1, its twist by $221$.
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 $0$.