Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+28444500x+635913800156\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+28444500xz^2+635913800156z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+455112000x+40698483210000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{11390526}{529}, \frac{40775713205}{12167}\right) \) | $12.921606214824228101118913102$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([261982098:40775713205:12167]\) | $12.921606214824228101118913102$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{45562104}{529}, \frac{326205754308}{12167}\right) \) | $12.921606214824228101118913102$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 452025 \) | = | $3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 41$ |
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| Minimal Discriminant: | $\Delta$ | = | $-176167813559185395010546875$ | = | $-1 \cdot 3^{9} \cdot 5^{8} \cdot 7^{6} \cdot 41^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{2813708206080}{194754273881} \) | = | $2^{18} \cdot 3^{3} \cdot 5 \cdot 41^{-7} \cdot 43^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.7156935952332691059736506273$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.84582069591512993514070077240$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1994714563919167$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.21290488468756$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $12.921606214824228101118913102$ |
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| Real period: | $\Omega$ | ≈ | $0.043550045220638723241344202779$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2\cdot3\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.7528384197465779803227518904 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.752838420 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.043550 \cdot 12.921606 \cdot 12}{1^2} \\ & \approx 6.752838420\end{aligned}$$
Modular invariants
Modular form 452025.2.a.dn
For more coefficients, see the Downloads section to the right.
| Modular degree: | 99792000 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $5$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $41$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
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 | $\ell$-adic index |
|---|---|---|---|
| $7$ | 7Ns | 7.28.0.1 | $28$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1722 = 2 \cdot 3 \cdot 7 \cdot 41 \), index $112$, genus $5$, and generators
$\left(\begin{array}{rr} 8 & 7 \\ 49 & 43 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 6 & 49 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1118 & 377 \\ 127 & 741 \end{array}\right),\left(\begin{array}{rr} 1709 & 14 \\ 1708 & 15 \end{array}\right),\left(\begin{array}{rr} 1159 & 8 \\ 809 & 171 \end{array}\right),\left(\begin{array}{rr} 211 & 14 \\ 1477 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1722])$ is a degree-$14282956800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1722\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 150675 = 3 \cdot 5^{2} \cdot 7^{2} \cdot 41 \) |
| $3$ | additive | $2$ | \( 50225 = 5^{2} \cdot 7^{2} \cdot 41 \) |
| $5$ | additive | $14$ | \( 18081 = 3^{2} \cdot 7^{2} \cdot 41 \) |
| $7$ | additive | $26$ | \( 225 = 3^{2} \cdot 5^{2} \) |
| $41$ | nonsplit multiplicative | $42$ | \( 11025 = 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 452025dn consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 9225a1, 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.