Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-858141075x+9528359805250\)
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(homogenize, simplify) |
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\(y^2z=x^3-858141075xz^2+9528359805250z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-858141075x+9528359805250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(181935435/12769, 635423834650/1442897)$ | $13.707541850371784171954846297$ | $\infty$ |
| $(18590, 0)$ | $0$ | $2$ |
Integral points
\( \left(18590, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 435600 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $1223056149523552278000000000$ | = | $2^{10} \cdot 3^{22} \cdot 5^{9} \cdot 11^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{3382175663521924}{59189241375} \) | = | $2^{2} \cdot 3^{-16} \cdot 5^{-3} \cdot 11^{-1} \cdot 94561^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.9941511541412415321587892782$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.86355576672433013594878843626$ |
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| $abc$ quality: | $Q$ | ≈ | $1.07876989573691$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.647083448630172$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $13.707541850371784171954846297$ |
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| Real period: | $\Omega$ | ≈ | $0.048619819391389441416125761604$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\cdot2^{2}\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $10.663331345039814034443024714 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.663331345 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.048620 \cdot 13.707542 \cdot 64}{2^2} \\ & \approx 10.663331345\end{aligned}$$
Modular invariants
Modular form 435600.2.a.si
For more coefficients, see the Downloads section to the right.
| Modular degree: | 283115520 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $3$ | $4$ | $I_{16}^{*}$ | additive | -1 | 2 | 22 | 16 |
| $5$ | $2$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $11$ | $4$ | $I_{1}^{*}$ | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 344 & 1317 \\ 435 & 878 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 499 & 498 \\ 618 & 835 \end{array}\right),\left(\begin{array}{rr} 476 & 879 \\ 537 & 434 \end{array}\right),\left(\begin{array}{rr} 1313 & 8 \\ 1312 & 9 \end{array}\right),\left(\begin{array}{rr} 439 & 0 \\ 0 & 1319 \end{array}\right),\left(\begin{array}{rr} 824 & 597 \\ 381 & 1232 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1314 & 1315 \end{array}\right)$.
The torsion field $K:=\Q(E[1320])$ is a degree-$9732096000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1320\Z)$.
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 | $2$ | \( 27225 = 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
| $3$ | additive | $8$ | \( 48400 = 2^{4} \cdot 5^{2} \cdot 11^{2} \) |
| $5$ | additive | $18$ | \( 17424 = 2^{4} \cdot 3^{2} \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 435600si
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1320n3, its twist by $-660$.
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.