Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-266682x-53007019\)
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(homogenize, simplify) |
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\(y^2z=x^3-266682xz^2-53007019z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-266682x-53007019\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-298, 25)$ | $2.5117102615890411423488497820$ | $\infty$ |
| $(-299, 0)$ | $0$ | $2$ |
Integral points
\( \left(-299, 0\right) \), \((-298,\pm 25)\), \((1222,\pm 38025)\)
Invariants
| Conductor: | $N$ | = | \( 425880 \) | = | $2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $29557447592400$ | = | $2^{4} \cdot 3^{7} \cdot 5^{2} \cdot 7 \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{37256083456}{525} \) | = | $2^{11} \cdot 3^{-1} \cdot 5^{-2} \cdot 7^{-1} \cdot 263^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7249751260035812944197545105$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.33785475724789035577702253590$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9605836908258326$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.7876384510364502$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.5117102615890411423488497820$ |
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| Real period: | $\Omega$ | ≈ | $0.21008812060917977497993166703$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.4428878139525273306931139611 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $4$ = $2^2$ (rounded) |
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BSD formula
$$\begin{aligned} 8.442887814 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.210088 \cdot 2.511710 \cdot 16}{2^2} \\ & \approx 8.442887814\end{aligned}$$
Modular invariants
Modular form 425880.2.a.eb
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2359296 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | -1 | 3 | 4 | 0 |
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $2$ | $I_0^{*}$ | 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 | 8.12.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 21840 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 14925 & 20176 \\ 2834 & 13131 \end{array}\right),\left(\begin{array}{rr} 11192 & 15119 \\ 481 & 5030 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21825 & 16 \\ 21824 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 20176 \\ 5460 & 5461 \end{array}\right),\left(\begin{array}{rr} 7216 & 11765 \\ 12675 & 8386 \end{array}\right),\left(\begin{array}{rr} 6719 & 0 \\ 0 & 21839 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 21742 & 21827 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 21836 & 21837 \end{array}\right),\left(\begin{array}{rr} 19501 & 20176 \\ 17888 & 16485 \end{array}\right)$.
The torsion field $K:=\Q(E[21840])$ is a degree-$155817722511360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/21840\Z)$.
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 | $2$ | \( 10647 = 3^{2} \cdot 7 \cdot 13^{2} \) |
| $3$ | additive | $8$ | \( 47320 = 2^{3} \cdot 5 \cdot 7 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 85176 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13^{2} \) |
| $7$ | split multiplicative | $8$ | \( 60840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 425880.eb
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 840.d4, its twist by $-39$.
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.