Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-185242575x+1057169337750\)
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(homogenize, simplify) |
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\(y^2z=x^3-185242575xz^2+1057169337750z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-185242575x+1057169337750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(605155/81, 218968750/729)$ | $7.8041368850728099343614058295$ | $\infty$ |
| $(-15870, 0)$ | $0$ | $2$ |
Integral points
\( \left(-15870, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 476100 \) | = | $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $-75986127905470312500000000$ | = | $-1 \cdot 2^{8} \cdot 3^{3} \cdot 5^{14} \cdot 23^{9} $ |
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| j-invariant: | $j$ | = | \( -\frac{3613864464}{390625} \) | = | $-1 \cdot 2^{4} \cdot 3^{3} \cdot 5^{-8} \cdot 7^{3} \cdot 29^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.7042055363670010585742237979$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.18888527433723569262485321611$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9981726076535594$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.2699730850833495$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.8041368850728099343614058295$ |
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| Real period: | $\Omega$ | ≈ | $0.059608082690508774200272964476$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 3\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.5822756412816354968871017678 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.582275641 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.059608 \cdot 7.804137 \cdot 48}{2^2} \\ & \approx 5.582275641\end{aligned}$$
Modular invariants
Modular form 476100.2.a.d
For more coefficients, see the Downloads section to the right.
| Modular degree: | 132268032 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $4$ | $I_{8}^{*}$ | additive | 1 | 2 | 14 | 8 |
| $23$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 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 | 8.12.0.34 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \), index $96$, genus $5$, and generators
$\left(\begin{array}{rr} 2656 & 5 \\ 531 & 16 \end{array}\right),\left(\begin{array}{rr} 4417 & 16 \\ 2216 & 129 \end{array}\right),\left(\begin{array}{rr} 1850 & 9 \\ 1811 & 5494 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 5396 & 5465 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2762 \\ 2 & 1385 \end{array}\right),\left(\begin{array}{rr} 2 & 4839 \\ 5507 & 1322 \end{array}\right),\left(\begin{array}{rr} 13 & 4 \\ 5380 & 5477 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 5456 & 5427 \end{array}\right),\left(\begin{array}{rr} 5505 & 16 \\ 5504 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[5520])$ is a degree-$1575820984320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5520\Z)$.
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 | $2$ | \( 1725 = 3 \cdot 5^{2} \cdot 23 \) |
| $3$ | additive | $6$ | \( 52900 = 2^{2} \cdot 5^{2} \cdot 23^{2} \) |
| $5$ | additive | $18$ | \( 19044 = 2^{2} \cdot 3^{2} \cdot 23^{2} \) |
| $23$ | additive | $156$ | \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 476100.d
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 95220.a2, its twist by $345$.
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.