Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-15150x-666875\)
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(homogenize, simplify) |
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\(y^2z=x^3-15150xz^2-666875z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-15150x-666875\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-70, 225)$ | $1.3378806516612476897411721988$ | $\infty$ |
| $(-55, 0)$ | $0$ | $2$ |
Integral points
\((-70,\pm 225)\), \( \left(-55, 0\right) \), \((141,\pm 14)\), \((350,\pm 6075)\)
Invariants
| Conductor: | $N$ | = | \( 412200 \) | = | $2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 229$ |
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| Discriminant: | $\Delta$ | = | $30424997250000$ | = | $2^{4} \cdot 3^{12} \cdot 5^{6} \cdot 229 $ |
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| j-invariant: | $j$ | = | \( \frac{2110056448}{166941} \) | = | $2^{11} \cdot 3^{-6} \cdot 101^{3} \cdot 229^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3308725360950281849747131642$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.25420162464272528449569982803$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8476422361397046$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.1317227825299474$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.3378806516612476897411721988$ |
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| Real period: | $\Omega$ | ≈ | $0.43248374269254316812343550510$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.6288930520511601785651526070 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.628893052 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.432484 \cdot 1.337881 \cdot 32}{2^2} \\ & \approx 4.628893052\end{aligned}$$
Modular invariants
Modular form 412200.2.a.d
For more coefficients, see the Downloads section to the right.
| Modular degree: | 835584 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | -1 | 3 | 4 | 0 |
| $3$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $5$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $229$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2748 = 2^{2} \cdot 3 \cdot 229 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 917 & 4 \\ 1834 & 9 \end{array}\right),\left(\begin{array}{rr} 2745 & 4 \\ 2744 & 5 \end{array}\right),\left(\begin{array}{rr} 689 & 2062 \\ 2060 & 687 \end{array}\right),\left(\begin{array}{rr} 470 & 1 \\ 1139 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[2748])$ is a degree-$1051390955520$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2748\Z)$.
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 | $2$ | \( 51525 = 3^{2} \cdot 5^{2} \cdot 229 \) |
| $3$ | additive | $2$ | \( 45800 = 2^{3} \cdot 5^{2} \cdot 229 \) |
| $5$ | additive | $14$ | \( 16488 = 2^{3} \cdot 3^{2} \cdot 229 \) |
| $229$ | nonsplit multiplicative | $230$ | \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 412200d
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 5496a1, its twist by $-15$.
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.