Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-26816075x+50880872250\)
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(homogenize, simplify) |
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\(y^2z=x^3-26816075xz^2+50880872250z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-26816075x+50880872250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2285, 39200)$ | $2.1417796475264530379761728827$ | $\infty$ |
| $(6695, 414050)$ | $0$ | $4$ |
Integral points
\((-4771,\pm 264992)\), \((-1755,\pm 304200)\), \((2285,\pm 39200)\), \( \left(3510, 0\right) \), \((6695,\pm 414050)\)
Invariants
| Conductor: | $N$ | = | \( 473200 \) | = | $2^{4} \cdot 5^{2} \cdot 7 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $115754468336037440000000$ | = | $2^{12} \cdot 5^{7} \cdot 7^{8} \cdot 13^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{6903498885921}{374712065} \) | = | $3^{3} \cdot 5^{-1} \cdot 7^{-8} \cdot 11^{3} \cdot 13^{-1} \cdot 577^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1814694974162314320330161642$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.40112868190846756728866065535$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0588031094756087$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.815628883729414$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.1417796475264530379761728827$ |
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| Real period: | $\Omega$ | ≈ | $0.10361062704867426521489111751$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 512 $ = $ 2^{2}\cdot2^{2}\cdot2^{3}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.1011562329697391662823896010 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.101156233 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.103611 \cdot 2.141780 \cdot 512}{4^2} \\ & \approx 7.101156233\end{aligned}$$
Modular invariants
Modular form 473200.2.a.em
For more coefficients, see the Downloads section to the right.
| Modular degree: | 33030144 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | -1 | 4 | 12 | 0 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $13$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 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 | 8.24.0.52 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7280 = 2^{4} \cdot 5 \cdot 7 \cdot 13 \), index $192$, genus $3$, and generators
$\left(\begin{array}{rr} 4161 & 16 \\ 4168 & 129 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6706 & 7277 \\ 6067 & 7260 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 8 & 65 \end{array}\right),\left(\begin{array}{rr} 3647 & 3650 \\ 1794 & 1783 \end{array}\right),\left(\begin{array}{rr} 7265 & 16 \\ 7264 & 17 \end{array}\right),\left(\begin{array}{rr} 7273 & 3624 \\ 4302 & 343 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1442 & 7277 \\ 2819 & 7260 \end{array}\right),\left(\begin{array}{rr} 15 & 166 \\ 6994 & 4115 \end{array}\right)$.
The torsion field $K:=\Q(E[7280])$ is a degree-$3246202552320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7280\Z)$.
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 | $2$ | \( 4225 = 5^{2} \cdot 13^{2} \) |
| $5$ | additive | $18$ | \( 18928 = 2^{4} \cdot 7 \cdot 13^{2} \) |
| $7$ | split multiplicative | $8$ | \( 67600 = 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 473200.em
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 455.a1, its twist by $-260$.
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.