Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-288x-1888\)
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(homogenize, simplify) |
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\(y^2z=x^3-288xz^2-1888z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-288x-1888\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(41, 235\right) \) | $3.1232755765464788282812852240$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([41:235:1]\) | $3.1232755765464788282812852240$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(41, 235\right) \) | $3.1232755765464788282812852240$ | $\infty$ |
Integral points
\((41,\pm 235)\)
\([41:\pm 235:1]\)
\((41,\pm 235)\)
Invariants
| Conductor: | $N$ | = | \( 8640 \) | = | $2^{6} \cdot 3^{3} \cdot 5$ |
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| Minimal Discriminant: | $\Delta$ | = | $-11059200$ | = | $-1 \cdot 2^{14} \cdot 3^{3} \cdot 5^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{5971968}{25} \) | = | $-1 \cdot 2^{13} \cdot 3^{6} \cdot 5^{-2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.20653594742291384050288311031$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.87678883539738310999936567396$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0904350906962674$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.156354144227532$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.1232755765464788282812852240$ |
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| Real period: | $\Omega$ | ≈ | $0.57931215615081900783856631662$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.6187030170046660157623024796 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.618703017 \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 0.579312 \cdot 3.123276 \cdot 2}{1^2} \\ & \approx 3.618703017\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2304 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | 1 | 6 | 14 | 0 |
| $3$ | $1$ | $II$ | additive | 1 | 3 | 3 | 0 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 | $\ell$-adic index |
|---|---|---|---|
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 24.16.0-6.b.1.1, level \( 24 = 2^{3} \cdot 3 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 22 & 21 \\ 23 & 20 \end{array}\right),\left(\begin{array}{rr} 20 & 21 \\ 21 & 23 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 11 & 0 \\ 0 & 23 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 19 & 6 \\ 18 & 7 \end{array}\right)$.
The torsion field $K:=\Q(E[24])$ is a degree-$4608$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24\Z)$.
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 | $2$ | \( 27 = 3^{3} \) |
| $3$ | additive | $2$ | \( 320 = 2^{6} \cdot 5 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 1728 = 2^{6} \cdot 3^{3} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 8640d
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 540c1, its twist by $-24$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 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 |
| $6$ | 6.2.11520000.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.4478976.2 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.1194393600000000.3 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | 12.0.20061226008576.9 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.13598118630822483386066534400000000.3 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.7312316880125952000000000000.1 | \(\Z/6\Z\) | 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 | nonsplit | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 1 | 1 | 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 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.