Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-108000x+13662000\)
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(homogenize, simplify) |
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\(y^2z=x^3-108000xz^2+13662000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-108000x+13662000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(145, 1025\right) \) | $3.1181079238671189523017797792$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([145:1025:1]\) | $3.1181079238671189523017797792$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(145, 1025\right) \) | $3.1181079238671189523017797792$ | $\infty$ |
Integral points
\((145,\pm 1025)\)
\([145:\pm 1025:1]\)
\((145,\pm 1025)\)
Invariants
| Conductor: | $N$ | = | \( 10800 \) | = | $2^{4} \cdot 3^{3} \cdot 5^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-11337408000000$ | = | $-1 \cdot 2^{12} \cdot 3^{11} \cdot 5^{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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5500140691840946959717991384$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.95491333220533468452478745018$ |
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| $abc$ quality: | $Q$ | ≈ | $1.23864473399791$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.994282095837806$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.1181079238671189523017797792$ |
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| Real period: | $\Omega$ | ≈ | $0.68421624586202311376135284228$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.2669201957219742201762781507 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.266920196 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.684216 \cdot 3.118108 \cdot 2}{1^2} \\ & \approx 4.266920196\end{aligned}$$
Modular invariants
\( q - q^{7} - 5 q^{13} + 7 q^{19} + O(q^{20}) \)
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For more coefficients, see the Downloads section to the right.
| Modular degree: | 31104 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | -1 | 4 | 12 | 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 $\ell$.
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 | $4$ | \( 400 = 2^{4} \cdot 5^{2} \) |
| $5$ | additive | $14$ | \( 432 = 2^{4} \cdot 3^{3} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 9 and 27.
Its isogeny class 10800cz
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 $60$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{15}) \) | \(\Z/3\Z\) | 2.2.60.1-81.1-a2 |
| $3$ | \(\Q(\sqrt[3]{2})\) | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.34992.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.472392000.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.6.157464000.1 | \(\Z/9\Z\) | not in database |
| $6$ | 6.2.69984000.1 | \(\Z/6\Z\) | 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 |
| $12$ | 12.0.2008387814976000000.1 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/7\Z\) | not in database |
| $12$ | 12.0.4897760256000000.3 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.18.1512608105754026853705216000000000.1 | \(\Z/27\Z\) | not in database |
| $18$ | 18.0.4918301009412067196928000000000.3 | \(\Z/6\Z\) | not in database |
| $18$ | 18.6.20239921849432375296000000000.2 | \(\Z/18\Z\) | 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 | add | add | add | ord | ss | ord | ss | ord | ss | ss | ord | ord | ss | ord | ss |
| $\lambda$-invariant(s) | - | - | - | 1 | 1,1 | 1 | 1,1 | 3 | 1,1 | 3,1 | 1 | 1 | 1,1 | 1 | 1,1 |
| $\mu$-invariant(s) | - | - | - | 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.