Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-135x-594\)
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(homogenize, simplify) |
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\(y^2z=x^3-135xz^2-594z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-135x-594\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-6, 0)$ | $0$ | $2$ |
Integral points
\( \left(-6, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 36 \) | = | $2^{2} \cdot 3^{2}$ |
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| Discriminant: | $\Delta$ | = | $5038848$ | = | $2^{8} \cdot 3^{9} $ |
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| j-invariant: | $j$ | = | \( 54000 \) | = | $2^{4} \cdot 3^{3} \cdot 5^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[\sqrt{-3}]\) (potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.080464438539665444737591499720$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.2055928983347136967536638423$ |
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| $abc$ quality: | $Q$ | ≈ | $1.027195810121916$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.347366602555891$ | |||
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: | $\Omega$ | ≈ | $1.4021821053254542611750190790$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.70109105266272713058750953952 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.701091053 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.402182 \cdot 1.000000 \cdot 2}{2^2} \\ & \approx 0.701091053\end{aligned}$$
Modular invariants
\( q - 4 q^{7} + 2 q^{13} + 8 q^{19} + O(q^{20}) \)
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For more coefficients, see the Downloads section to the right.
| Modular degree: | 6 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 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 |
|---|---|---|
| $3$ | 3B.1.2 | 27.648.18.4 |
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 | $2$ | \( 3 \) |
| $3$ | additive | $2$ | \( 4 = 2^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 36.a
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 36.a2, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.2.12.1-36.1-a5 |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/6\Z\) | 2.0.3.1-144.1-CMa2 |
| $3$ | 3.1.108.1 | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.432.1 | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\zeta_{12})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.34992.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.559872.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.2985984.1 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.4.191102976.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $12$ | 12.0.84687918336.1 | \(\Z/42\Z\) | not in database |
| $12$ | 12.0.313456656384.1 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.940369969152.1 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $16$ | 16.0.36520347436056576.1 | \(\Z/4\Z \oplus \Z/12\Z\) | not in database |
| $16$ | 16.0.36520347436056576.2 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $18$ | 18.0.2529990231179046912.1 | \(\Z/3\Z \oplus \Z/18\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
| $p$ | 2 | 3 |
|---|---|---|
| Reduction type | add | add |
| $\lambda$-invariant(s) | - | - |
| $\mu$-invariant(s) | - | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
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 $0$.