Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-2700x-67500\)
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(homogenize, simplify) |
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\(y^2z=x^3-2700xz^2-67500z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2700x-67500\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 5400 \) | = | $2^{3} \cdot 3^{3} \cdot 5^{2}$ |
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| Discriminant: | $\Delta$ | = | $-708588000000$ | = | $-1 \cdot 2^{8} \cdot 3^{11} \cdot 5^{6} $ |
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| j-invariant: | $j$ | = | \( -3072 \) | = | $-1 \cdot 2^{10} \cdot 3$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.98642957195172261981653614832$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.2874487692510583242076397331$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9463948908766787$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.161306665392271$ | |||
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: | $\Omega$ | ≈ | $0.32575809570310852598845956989$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.6060647656248682079076765591 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.606064766 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.325758 \cdot 1.000000 \cdot 8}{1^2} \\ & \approx 2.606064766\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5760 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{1}^{*}$ | additive | 1 | 3 | 8 | 0 |
| $3$ | $1$ | $II^{*}$ | additive | -1 | 3 | 11 | 0 |
| $5$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2G | 4.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 12.8.0.c.1, level \( 12 = 2^{2} \cdot 3 \), index $8$, genus $0$, and generators
$\left(\begin{array}{rr} 9 & 4 \\ 8 & 5 \end{array}\right),\left(\begin{array}{rr} 4 & 9 \\ 5 & 1 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 5 & 3 \end{array}\right),\left(\begin{array}{rr} 8 & 3 \\ 1 & 4 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[12])$ is a degree-$576$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/12\Z)$.
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 | $2$ | \( 675 = 3^{3} \cdot 5^{2} \) |
| $3$ | additive | $4$ | \( 200 = 2^{3} \cdot 5^{2} \) |
| $5$ | additive | $14$ | \( 216 = 2^{3} \cdot 3^{3} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 5400n consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 216a1, its twist by $-15$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.108.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.34992.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.113374080000.6 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | add | ord | ord | ord | ord | ord | ord | ss | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | - | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0,0 | 0 | 0 |
| $\mu$-invariant(s) | - | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0,0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.