Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-36864x-2317680\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-36864xz^2-2317680z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2986011x-1680630714\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-96, 588)$ | $1.2900751157287369256185472214$ | $\infty$ |
| $(-145, 0)$ | $0$ | $2$ |
Integral points
\( \left(-145, 0\right) \), \((-96,\pm 588)\), \((255,\pm 2220)\), \((384,\pm 6348)\)
Invariants
| Conductor: | $N$ | = | \( 27048 \) | = | $2^{3} \cdot 3 \cdot 7^{2} \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $910255485791232$ | = | $2^{10} \cdot 3^{3} \cdot 7^{6} \cdot 23^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{45989074372}{7555707} \) | = | $2^{2} \cdot 3^{-3} \cdot 23^{-4} \cdot 37^{3} \cdot 61^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5920261083765146069693130869$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.041448383382236863235609947296$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9626262468353085$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.2290097657512$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.2900751157287369256185472214$ |
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| Real period: | $\Omega$ | ≈ | $0.34838557822549112880203963216$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2\cdot3\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.3933227817696807895198646069 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.393322782 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.348386 \cdot 1.290075 \cdot 48}{2^2} \\ & \approx 5.393322782\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 110592 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | -1 | 3 | 10 | 0 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $23$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 211 & 210 \\ 2842 & 2563 \end{array}\right),\left(\begin{array}{rr} 1103 & 0 \\ 0 & 3863 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 3857 & 8 \\ 3856 & 9 \end{array}\right),\left(\begin{array}{rr} 2857 & 560 \\ 2044 & 2241 \end{array}\right),\left(\begin{array}{rr} 2948 & 553 \\ 3703 & 2766 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 3858 & 3859 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 624 & 2009 \\ 3409 & 1800 \end{array}\right)$.
The torsion field $K:=\Q(E[3864])$ is a degree-$827306016768$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3864\Z)$.
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 | $2$ | \( 147 = 3 \cdot 7^{2} \) |
| $3$ | split multiplicative | $4$ | \( 9016 = 2^{3} \cdot 7^{2} \cdot 23 \) |
| $7$ | additive | $26$ | \( 552 = 2^{3} \cdot 3 \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 27048.n
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 552.c1, its twist by $-7$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-21}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{-7})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.7169347584.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.24768856756224.5 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.32100438356066304.10 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.18576642567168.1 | \(\Z/6\Z\) | not in database |
| $16$ | 16.0.51399544780206637056.9 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | split | ord | add | ord | ord | ord | ss | split | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | 1 | - | 1 | 1 | 1 | 1,1 | 2 | 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 |
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.