Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+40500x+486000\)
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(homogenize, simplify) |
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\(y^2z=x^3+40500xz^2+486000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+40500x+486000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4885/9, 364625/27)$ | $7.0742969146472710292144401768$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 129600 \) | = | $2^{6} \cdot 3^{4} \cdot 5^{2}$ |
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| Discriminant: | $\Delta$ | = | $-4353564672000000$ | = | $-1 \cdot 2^{19} \cdot 3^{12} \cdot 5^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{3375}{2} \) | = | $2^{-1} \cdot 3^{3} \cdot 5^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6901588757410065264020143463$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.2528931399840713164194587394$ |
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| $abc$ quality: | $Q$ | ≈ | $1.4265653296335434$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.690112726507918$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.0742969146472710292144401768$ |
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| Real period: | $\Omega$ | ≈ | $0.26625889918861606342171128003$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.5343780361096215724414391153 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.534378036 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.266259 \cdot 7.074297 \cdot 4}{1^2} \\ & \approx 7.534378036\end{aligned}$$
Modular invariants
Modular form 129600.2.a.cy
For more coefficients, see the Downloads section to the right.
| Modular degree: | 497664 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{9}^{*}$ | additive | 1 | 6 | 19 | 1 |
| $3$ | $1$ | $II^{*}$ | additive | 1 | 4 | 12 | 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 | 8.2.0.1 |
| $3$ | 3B | 3.4.0.1 |
| $7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \), index $768$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 1470 & 1 \end{array}\right),\left(\begin{array}{rr} 281 & 1400 \\ 1120 & 1401 \end{array}\right),\left(\begin{array}{rr} 1 & 930 \\ 1050 & 1261 \end{array}\right),\left(\begin{array}{rr} 2311 & 1050 \\ 735 & 211 \end{array}\right),\left(\begin{array}{rr} 503 & 0 \\ 0 & 2519 \end{array}\right),\left(\begin{array}{rr} 1479 & 350 \\ 560 & 879 \end{array}\right),\left(\begin{array}{rr} 419 & 1470 \\ 315 & 1469 \end{array}\right),\left(\begin{array}{rr} 1 & 720 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1680 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1050 \\ 0 & 1081 \end{array}\right),\left(\begin{array}{rr} 1471 & 1290 \\ 420 & 2311 \end{array}\right),\left(\begin{array}{rr} 505 & 2016 \\ 504 & 505 \end{array}\right),\left(\begin{array}{rr} 526 & 825 \\ 2205 & 1681 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1680 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2016 & 1 \end{array}\right),\left(\begin{array}{rr} 841 & 1680 \\ 840 & 1681 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 420 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[2520])$ is a degree-$7524679680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2520\Z)$.
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 | $4$ | \( 2025 = 3^{4} \cdot 5^{2} \) |
| $3$ | additive | $2$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
| $5$ | additive | $14$ | \( 5184 = 2^{6} \cdot 3^{4} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 7 and 21.
Its isogeny class 129600.cy
consists of 4 curves linked by isogenies of
degrees dividing 21.
Twists
The minimal quadratic twist of this elliptic curve is 162.b4, its twist by $-120$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-30}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.648.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.3359232.4 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.186624000.3 | \(\Z/3\Z\) | not in database |
| $6$ | 6.6.419904000.1 | \(\Z/7\Z\) | not in database |
| $6$ | 6.0.5038848000.13 | \(\Z/6\Z\) | 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 |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.1586874322944000000.2 | \(\Z/21\Z\) | not in database |
| $18$ | 18.0.1376809511370776442839236608000000000.2 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.55268479930183339474944000000000.3 | \(\Z/6\Z\) | not in database |
| $18$ | 18.6.4738381338321616896000000000.1 | \(\Z/14\Z\) | not in database |
| $18$ | 18.6.3454279995636458717184000000000.2 | \(\Z/21\Z\) | 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 | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | - | - | 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.