Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-255481707x+1571763139994\)
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(homogenize, simplify) |
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\(y^2z=x^3-255481707xz^2+1571763139994z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-255481707x+1571763139994\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(9413, 30800)$ | $2.0417984447836971725984427567$ | $\infty$ |
| $(8503, 119070)$ | $0$ | $4$ |
Integral points
\((-8507,\pm 1769040)\), \((8503,\pm 119070)\), \( \left(9238, 0\right) \), \((9413,\pm 30800)\), \((12325,\pm 543312)\)
Invariants
| Conductor: | $N$ | = | \( 95760 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $3576384341396213760000$ | = | $2^{15} \cdot 3^{13} \cdot 5^{4} \cdot 7^{8} \cdot 19 $ |
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| j-invariant: | $j$ | = | \( \frac{617611911727813844500009}{1197723879765000} \) | = | $2^{-3} \cdot 3^{-7} \cdot 5^{-4} \cdot 7^{-8} \cdot 13^{3} \cdot 19^{-1} \cdot 37^{3} \cdot 47^{3} \cdot 3767^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.3891394649675117563085525906$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.1466861400735116011936978507$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0155441932694813$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.076025761141981$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.0417984447836971725984427567$ |
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| Real period: | $\Omega$ | ≈ | $0.12056776278199644018308407114$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 512 $ = $ 2^{2}\cdot2^{2}\cdot2^{2}\cdot2^{3}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.8776022572585618425001759550 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.877602257 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.120568 \cdot 2.041798 \cdot 512}{4^2} \\ & \approx 7.877602257\end{aligned}$$
Modular invariants
Modular form 95760.2.a.fl
For more coefficients, see the Downloads section to the right.
| Modular degree: | 12386304 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{7}^{*}$ | additive | -1 | 4 | 15 | 3 |
| $3$ | $4$ | $I_{7}^{*}$ | additive | -1 | 2 | 13 | 7 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $7$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $19$ | $1$ | $I_{1}$ | nonsplit 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 | 4.12.0.7 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 277 & 284 \\ 238 & 1419 \end{array}\right),\left(\begin{array}{rr} 288 & 1433 \\ 1987 & 1974 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 752 & 2277 \\ 755 & 2278 \end{array}\right),\left(\begin{array}{rr} 2273 & 8 \\ 2272 & 9 \end{array}\right),\left(\begin{array}{rr} 457 & 8 \\ 1828 & 33 \end{array}\right),\left(\begin{array}{rr} 844 & 1 \\ 863 & 6 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 2274 & 2275 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[2280])$ is a degree-$90773913600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2280\Z)$.
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 | $4$ | \( 171 = 3^{2} \cdot 19 \) |
| $3$ | additive | $8$ | \( 10640 = 2^{4} \cdot 5 \cdot 7 \cdot 19 \) |
| $5$ | split multiplicative | $6$ | \( 19152 = 2^{4} \cdot 3^{2} \cdot 7 \cdot 19 \) |
| $7$ | split multiplicative | $8$ | \( 13680 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 95760ff
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 3990l4, its twist by $12$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{114}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.102600.1 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.243210263040000.2 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | split | split | ord | ord | ord | nonsplit | ord | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | - | 2 | 2 | 1 | 1 | 3 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1,3 |
| $\mu$-invariant(s) | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.