Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+1259697525x+4490719240250\)
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(homogenize, simplify) |
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\(y^2z=x^3+1259697525xz^2+4490719240250z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+1259697525x+4490719240250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4231834, 8705790702)$ | $6.8663310670634071973872149165$ | $\infty$ |
| $(-3530, 0)$ | $0$ | $2$ |
Integral points
\( \left(-3530, 0\right) \), \((4231834,\pm 8705790702)\)
Invariants
| Conductor: | $N$ | = | \( 428400 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-136643839559574653472768000000$ | = | $-1 \cdot 2^{18} \cdot 3^{10} \cdot 5^{6} \cdot 7^{16} \cdot 17 $ |
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| j-invariant: | $j$ | = | \( \frac{4738217997934888496063}{2928751705237796928} \) | = | $2^{-6} \cdot 3^{-4} \cdot 7^{-16} \cdot 17^{-1} \cdot 47^{3} \cdot 191^{3} \cdot 1871^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.2797276601540954502414254669$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.2325553790430451078261910604$ |
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| $abc$ quality: | $Q$ | ≈ | $1.06742381122777$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.743144016485853$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.8663310670634071973872149165$ |
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| Real period: | $\Omega$ | ≈ | $0.020254729454871682313180763625$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 256 $ = $ 2\cdot2^{2}\cdot2\cdot2^{4}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.9008433991007809532234998791 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.900843399 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.020255 \cdot 6.866331 \cdot 256}{2^2} \\ & \approx 8.900843399\end{aligned}$$
Modular invariants
Modular form 428400.2.a.nu
For more coefficients, see the Downloads section to the right.
| Modular degree: | 377487360 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{10}^{*}$ | additive | -1 | 4 | 18 | 6 |
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $16$ | $I_{16}$ | split multiplicative | -1 | 1 | 16 | 16 |
| $17$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 | 16.48.0.120 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 4065 & 16 \\ 4064 & 17 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 4076 & 4077 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2461 & 1920 \\ 60 & 121 \end{array}\right),\left(\begin{array}{rr} 2719 & 0 \\ 0 & 4079 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 3982 & 4067 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1454 & 1965 \\ 2865 & 2834 \end{array}\right),\left(\begin{array}{rr} 3263 & 0 \\ 0 & 4079 \end{array}\right),\left(\begin{array}{rr} 3856 & 1365 \\ 1395 & 4066 \end{array}\right)$.
The torsion field $K:=\Q(E[4080])$ is a degree-$231022264320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4080\Z)$.
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 | $2$ | \( 3825 = 3^{2} \cdot 5^{2} \cdot 17 \) |
| $3$ | additive | $8$ | \( 47600 = 2^{4} \cdot 5^{2} \cdot 7 \cdot 17 \) |
| $5$ | additive | $14$ | \( 17136 = 2^{4} \cdot 3^{2} \cdot 7 \cdot 17 \) |
| $7$ | split multiplicative | $8$ | \( 61200 = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 25200 = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 428400nu
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 714g4, its twist by $60$.
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.