Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2+12344608592x+461681261413312\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z+12344608592xz^2+461681261413312z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+999913295925x+336568639310192250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-478643099347779373752/15010330262322409, 10975194263217112222038758000000/1839015424164353557577077)$ | $38.046479156453116112465449908$ | $\infty$ |
| $(-34168, 0)$ | $0$ | $2$ |
Integral points
\( \left(-34168, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 433200 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $-212478232230351456000000000000000$ | = | $-1 \cdot 2^{20} \cdot 3 \cdot 5^{15} \cdot 19^{12} $ |
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| j-invariant: | $j$ | = | \( \frac{69096190760262356111}{70568821500000000} \) | = | $2^{-8} \cdot 3^{-1} \cdot 5^{-9} \cdot 19^{-6} \cdot 29^{3} \cdot 141499^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.8898714571658635578529235084$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.9197858308056478311307980044$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0444928492259786$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.265763313229332$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $38.046479156453116112465449908$ |
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| Real period: | $\Omega$ | ≈ | $0.011728528546693718639617065552$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot1\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.1396674702023667871733963379 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.139667470 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.011729 \cdot 38.046479 \cdot 64}{2^2} \\ & \approx 7.139667470\end{aligned}$$
Modular invariants
Modular form 433200.2.a.dw
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1791590400 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{12}^{*}$ | additive | -1 | 4 | 20 | 8 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $4$ | $I_{9}^{*}$ | additive | 1 | 2 | 15 | 9 |
| $19$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 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 \( 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1041 & 662 \\ 454 & 79 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1130 & 1137 \\ 939 & 8 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 1090 & 1131 \end{array}\right),\left(\begin{array}{rr} 359 & 1128 \\ 1014 & 1067 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1129 & 12 \\ 1128 & 13 \end{array}\right),\left(\begin{array}{rr} 770 & 3 \\ 733 & 1132 \end{array}\right)$.
The torsion field $K:=\Q(E[1140])$ is a degree-$2836684800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1140\Z)$.
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 | $2$ | \( 27075 = 3 \cdot 5^{2} \cdot 19^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 144400 = 2^{4} \cdot 5^{2} \cdot 19^{2} \) |
| $5$ | additive | $18$ | \( 17328 = 2^{4} \cdot 3 \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 433200.dw
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 570.k4, its twist by $380$.
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.