Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+7350x-171500\)
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(homogenize, simplify) |
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\(y^2z=x^3+7350xz^2-171500z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+7350x-171500\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 12700800 \) | = | $2^{7} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-38118276000000$ | = | $-1 \cdot 2^{8} \cdot 3^{4} \cdot 5^{6} \cdot 7^{6} $ |
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| j-invariant: | $j$ | = | \( 1152 \) | = | $2^{7} \cdot 3^{2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2935044050030829774981878243$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.3124718423376239657647713740$ |
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| $abc$ quality: | $Q$ | ≈ | $0.6131471927654584$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.342762031987346$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.35707009594240986167136710185$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.4282803837696394466854684074 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.428280384 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.357070 \cdot 1.000000 \cdot 4}{1^2} \\ & \approx 1.428280384\end{aligned}$$
Modular invariants
Modular form 12700800.2.a.db
For more coefficients, see the Downloads section to the right.
| Modular degree: | 29859840 |
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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 | 7 | 8 | 0 |
| $3$ | $1$ | $II$ | additive | 1 | 4 | 4 | 0 |
| $5$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2G | 4.2.0.1 | $2$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 280 = 2^{3} \cdot 5 \cdot 7 \), index $4$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 141 & 140 \\ 210 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 277 & 4 \\ 276 & 5 \end{array}\right),\left(\begin{array}{rr} 39 & 0 \\ 0 & 279 \end{array}\right),\left(\begin{array}{rr} 2 & 3 \\ 275 & 273 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 223 & 0 \\ 0 & 279 \end{array}\right),\left(\begin{array}{rr} 141 & 105 \\ 210 & 141 \end{array}\right)$.
The torsion field $K:=\Q(E[280])$ is a degree-$371589120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/280\Z)$.
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 | $2$ | \( 99225 = 3^{4} \cdot 5^{2} \cdot 7^{2} \) |
| $3$ | additive | $8$ | \( 156800 = 2^{7} \cdot 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $14$ | \( 508032 = 2^{7} \cdot 3^{4} \cdot 7^{2} \) |
| $7$ | additive | $26$ | \( 259200 = 2^{7} \cdot 3^{4} \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 12700800.db consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 10368.a1, its twist by $-35$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.