Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-330750x+73219781\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-330750xz^2+73219781z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5292000x+4686066000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{2965}{9}, \frac{2924}{27}\right) \) | $4.8544822096501563321357874527$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([8895:2924:27]\) | $4.8544822096501563321357874527$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{11860}{9}, \frac{23500}{27}\right) \) | $4.8544822096501563321357874527$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 33075 \) | = | $3^{3} \cdot 5^{2} \cdot 7^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-325643240671875$ | = | $-1 \cdot 3^{11} \cdot 5^{6} \cdot 7^{6} $ |
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| j-invariant: | $j$ | = | \( -12288000 \) | = | $-1 \cdot 2^{15} \cdot 3 \cdot 5^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-27})/2]\) (potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8298219631518060391072433886$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.95491333220533468452478745025$ |
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| $abc$ quality: | $Q$ | ≈ | $1.23864473399791$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.779795340717871$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.8544822096501563321357874527$ |
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| Real period: | $\Omega$ | ≈ | $0.51721886558318283134212234369$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.0216595629379931698071510650 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.021659563 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.517219 \cdot 4.854482 \cdot 2}{1^2} \\ & \approx 5.021659563\end{aligned}$$
Modular invariants
\( q - 2 q^{4} + 5 q^{13} + 4 q^{16} + 7 q^{19} + O(q^{20}) \)
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For more coefficients, see the Downloads section to the right.
| Modular degree: | 163296 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $II^{*}$ | additive | -1 | 3 | 11 | 0 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
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 |
|---|---|---|---|
| $3$ | additive | $4$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $14$ | \( 1323 = 3^{3} \cdot 7^{2} \) |
| $7$ | additive | $26$ | \( 675 = 3^{3} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 9 and 27.
Its isogeny class 33075bn
consists of 4 curves linked by isogenies of
degrees dividing 27.
Twists
The minimal quadratic twist of this elliptic curve is 27a4, its twist by $105$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/3\Z\) | not in database |
| $3$ | \(\Q(\sqrt[3]{2})\) | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.34992.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.2531725875.3 | \(\Z/3\Z\) | not in database |
| $6$ | 6.6.843908625.1 | \(\Z/9\Z\) | not in database |
| $6$ | 6.2.1500282000.3 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | 12.0.57686723155300640625.1 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/7\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.18.232846042803239587104289170509765625.1 | \(\Z/27\Z\) | not in database |
| $18$ | 18.0.48454879404667446465435576000000000.4 | \(\Z/6\Z\) | not in database |
| $18$ | 18.6.199402795903981261174632000000000.3 | \(\Z/18\Z\) | 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 | ss | add | add | add | ss | ord | ss | ord | ss | ss | ord | ord | ss | ord | ss |
| $\lambda$-invariant(s) | 2,7 | - | - | - | 1,1 | 1 | 1,3 | 1 | 1,1 | 1,1 | 1 | 1 | 1,1 | 3 | 1,1 |
| $\mu$-invariant(s) | 0,0 | - | - | - | 0,0 | 0 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.