Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-6732300x-6692722000\)
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(homogenize, simplify) |
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\(y^2z=x^3-6732300xz^2-6692722000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6732300x-6692722000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-1435, 3625\right) \) | $1.8127499504836295358517109707$ | $\infty$ |
| \( \left(-1580, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-1435:3625:1]\) | $1.8127499504836295358517109707$ | $\infty$ |
| \([-1580:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-1435, 3625\right) \) | $1.8127499504836295358517109707$ | $\infty$ |
| \( \left(-1580, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-1580, 0\right) \), \((-1435,\pm 3625)\), \((3640,\pm 130500)\)
\([-1580:0:1]\), \([-1435:\pm 3625:1]\), \([3640:\pm 130500:1]\)
\( \left(-1580, 0\right) \), \((-1435,\pm 3625)\), \((3640,\pm 130500)\)
Invariants
| Conductor: | $N$ | = | \( 417600 \) | = | $2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 29$ |
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| Minimal Discriminant: | $\Delta$ | = | $178194072614400000000$ | = | $2^{15} \cdot 3^{9} \cdot 5^{8} \cdot 29^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{90410028096968}{477414675} \) | = | $2^{3} \cdot 3^{-3} \cdot 5^{-2} \cdot 29^{-4} \cdot 22441^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7305874014526447997856547872$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.51012832520160813001611235030$ |
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| $abc$ quality: | $Q$ | ≈ | $0.944842546038884$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.541772263548344$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.8127499504836295358517109707$ |
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| Real period: | $\Omega$ | ≈ | $0.093755499219892256521921819405$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2\cdot2^{2}\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.4385688501896849787764126098 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.438568850 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.093755 \cdot 1.812750 \cdot 128}{2^2} \\ & \approx 5.438568850\end{aligned}$$
Modular invariants
Modular form 417600.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 18874368 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{5}^{*}$ | additive | 1 | 6 | 15 | 0 |
| $3$ | $4$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $29$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
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 |
|---|---|---|---|
| $2$ | 2B | 4.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3480 = 2^{3} \cdot 3 \cdot 5 \cdot 29 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 2641 & 1400 \\ 820 & 2121 \end{array}\right),\left(\begin{array}{rr} 1824 & 775 \\ 965 & 1674 \end{array}\right),\left(\begin{array}{rr} 2783 & 0 \\ 0 & 3479 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 3474 & 3475 \end{array}\right),\left(\begin{array}{rr} 3244 & 695 \\ 905 & 3474 \end{array}\right),\left(\begin{array}{rr} 1656 & 2705 \\ 3385 & 2676 \end{array}\right),\left(\begin{array}{rr} 3473 & 8 \\ 3472 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[3480])$ is a degree-$502883942400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3480\Z)$.
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 | $4$ | \( 225 = 3^{2} \cdot 5^{2} \) |
| $3$ | additive | $6$ | \( 46400 = 2^{6} \cdot 5^{2} \cdot 29 \) |
| $5$ | additive | $18$ | \( 16704 = 2^{6} \cdot 3^{2} \cdot 29 \) |
| $29$ | split multiplicative | $30$ | \( 14400 = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 417600h
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 13920h3, its twist by $-120$.
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.