Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-47416875x-125674683750\)
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(homogenize, simplify) |
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\(y^2z=x^3-47416875xz^2-125674683750z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-47416875x-125674683750\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 435600 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-9260303659200000000$ | = | $-1 \cdot 2^{12} \cdot 3^{3} \cdot 5^{8} \cdot 11^{8} $ |
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| j-invariant: | $j$ | = | \( -1273201875 \) | = | $-1 \cdot 3^{3} \cdot 5^{4} \cdot 11 \cdot 19^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9460614989527996402996338447$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.69329421059582037107487819345$ |
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| $abc$ quality: | $Q$ | ≈ | $1.148768405405738$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.978025476597732$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.028766452416969042384829086758$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 72 $ = $ 2^{2}\cdot2\cdot3\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.0711845740217710517076942466 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.071184574 \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 0.028766 \cdot 1.000000 \cdot 72}{1^2} \\ & \approx 2.071184574\end{aligned}$$
Modular invariants
Modular form 435600.2.a.ll
For more coefficients, see the Downloads section to the right.
| Modular degree: | 22302720 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | -1 | 4 | 12 | 0 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $11$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 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 |
|---|---|---|---|
| $5$ | 5Nn | 5.10.0.1 | $10$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 30.120.7.e.1, level \( 30 = 2 \cdot 3 \cdot 5 \), index $120$, genus $7$, and generators
$\left(\begin{array}{rr} 1 & 12 \\ 12 & 25 \end{array}\right),\left(\begin{array}{rr} 3 & 13 \\ 23 & 2 \end{array}\right),\left(\begin{array}{rr} 11 & 25 \\ 5 & 6 \end{array}\right),\left(\begin{array}{rr} 5 & 18 \\ 18 & 11 \end{array}\right),\left(\begin{array}{rr} 28 & 25 \\ 11 & 22 \end{array}\right),\left(\begin{array}{rr} 29 & 0 \\ 0 & 29 \end{array}\right),\left(\begin{array}{rr} 29 & 10 \\ 10 & 19 \end{array}\right)$.
The torsion field $K:=\Q(E[30])$ is a degree-$1152$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/30\Z)$.
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 | $2$ | \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \) |
| $3$ | additive | $6$ | \( 48400 = 2^{4} \cdot 5^{2} \cdot 11^{2} \) |
| $5$ | additive | $10$ | \( 17424 = 2^{4} \cdot 3^{2} \cdot 11^{2} \) |
| $11$ | additive | $52$ | \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 435600ll consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 27225j1, its twist by $220$.
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 $0$.