Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-17158575x-14860732250\)
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(homogenize, simplify) |
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\(y^2z=x^3-17158575xz^2-14860732250z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-17158575x-14860732250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-2170, 110250)$ | $2.4915339522112003679369674811$ | $\infty$ |
| $(-910, 0)$ | $0$ | $2$ |
Integral points
\((-2170,\pm 110250)\), \( \left(-910, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 485100 \) | = | $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $227909872627321500000000$ | = | $2^{8} \cdot 3^{7} \cdot 5^{9} \cdot 7^{6} \cdot 11^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{1628514404944}{664335375} \) | = | $2^{4} \cdot 3^{-1} \cdot 5^{-3} \cdot 7^{3} \cdot 11^{-6} \cdot 23^{3} \cdot 29^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1795879719221490934751404942$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.39050967647009053497964042310$ |
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| $abc$ quality: | $Q$ | ≈ | $0.963342172697873$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.704178874168032$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.4915339522112003679369674811$ |
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| Real period: | $\Omega$ | ≈ | $0.076873414880172871716227578169$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 192 $ = $ 1\cdot2\cdot2^{2}\cdot2^{2}\cdot( 2 \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $9.1935707134256839208440887263 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.193570713 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.076873 \cdot 2.491534 \cdot 192}{2^2} \\ & \approx 9.193570713\end{aligned}$$
Modular invariants
Modular form 485100.2.a.eu
For more coefficients, see the Downloads section to the right.
| Modular degree: | 47775744 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $4$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
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 | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 3730 & 3297 \\ 1127 & 3968 \end{array}\right),\left(\begin{array}{rr} 4609 & 12 \\ 4608 & 13 \end{array}\right),\left(\begin{array}{rr} 650 & 3297 \\ 819 & 3968 \end{array}\right),\left(\begin{array}{rr} 3959 & 0 \\ 0 & 4619 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1926 & 1057 \\ 1925 & 386 \end{array}\right),\left(\begin{array}{rr} 2521 & 672 \\ 3906 & 4033 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 4570 & 4611 \end{array}\right)$.
The torsion field $K:=\Q(E[4620])$ is a degree-$613122048000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4620\Z)$.
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 | $2$ | \( 11025 = 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
| $3$ | additive | $8$ | \( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $18$ | \( 19404 = 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11 \) |
| $7$ | additive | $26$ | \( 9900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 44100 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 485100eu
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 660c4, its twist by $105$.
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.