Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-6085800x-6371568000\)
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(homogenize, simplify) |
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\(y^2z=x^3-6085800xz^2-6371568000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6085800x-6371568000\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 25401600 \) | = | $2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $-3112280998848000000000$ | = | $-1 \cdot 2^{15} \cdot 3^{10} \cdot 5^{9} \cdot 7^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{7008192}{875} \) | = | $-1 \cdot 2^{6} \cdot 3^{2} \cdot 5^{-3} \cdot 7^{-1} \cdot 23^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8584476545244641181769228534$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.70117059247693243461037770086$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7842206106422358$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.44118814370729$ | |||
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: | $\Omega$ | ≈ | $0.047727319970089421475906988646$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2\cdot3\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.2909113585642922308435354550 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.290911359 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.047727 \cdot 1.000000 \cdot 48}{1^2} \\ & \approx 2.290911359\end{aligned}$$
Modular invariants
Modular form 25401600.2.a.f
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1783627776 |
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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 | 8 | 15 | 0 |
| $3$ | $3$ | $IV^{*}$ | additive | 1 | 4 | 10 | 0 |
| $5$ | $4$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $7$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 280 = 2^{3} \cdot 5 \cdot 7 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 279 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 141 & 2 \\ 141 & 3 \end{array}\right),\left(\begin{array}{rr} 71 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 279 & 2 \\ 278 & 3 \end{array}\right),\left(\begin{array}{rr} 241 & 2 \\ 241 & 3 \end{array}\right),\left(\begin{array}{rr} 57 & 2 \\ 57 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[280])$ is a degree-$743178240$ 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 | $4$ | \( 99225 = 3^{4} \cdot 5^{2} \cdot 7^{2} \) |
| $3$ | additive | $4$ | \( 4480 = 2^{7} \cdot 5 \cdot 7 \) |
| $5$ | additive | $18$ | \( 1016064 = 2^{8} \cdot 3^{4} \cdot 7^{2} \) |
| $7$ | additive | $32$ | \( 518400 = 2^{8} \cdot 3^{4} \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 25401600.f consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 725760.i1, its twist by $140$.
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 $0$.