Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-8327937x-9253043361\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-8327937xz^2-9253043361z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-674562924x-6743444921424\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 55488 \) | = | $2^{6} \cdot 3 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-16364077056$ | = | $-1 \cdot 2^{21} \cdot 3^{3} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{843137281012581793}{216} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-3} \cdot 7^{3} \cdot 17 \cdot 73^{3} \cdot 719^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2422659195037536145094242817$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.73034292465446630367532032987$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0840070424605754$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.439345117888885$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.044436024432556159479554486881$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 2\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.3995453193580326118959422916 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.399545319 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.044436 \cdot 1.000000 \cdot 6}{1^2} \\ & \approx 2.399545319\end{aligned}$$
Modular invariants
Modular form 55488.2.a.cn
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1306368 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{11}^{*}$ | additive | 1 | 6 | 21 | 3 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $17$ | $1$ | $II$ | additive | 1 | 2 | 2 | 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 |
|---|---|---|
| $3$ | 3B | 9.36.0.9 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \), index $144$, genus $2$, and generators
$\left(\begin{array}{rr} 611 & 1206 \\ 603 & 1061 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 12 & 217 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 307 & 630 \\ 315 & 1217 \end{array}\right),\left(\begin{array}{rr} 17 & 18 \\ 864 & 305 \end{array}\right),\left(\begin{array}{rr} 1207 & 18 \\ 1206 & 19 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 13 & 12 \\ 1160 & 1165 \end{array}\right),\left(\begin{array}{rr} 919 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1224])$ is a degree-$3248750592$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1224\Z)$.
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 | $4$ | \( 867 = 3 \cdot 17^{2} \) |
| $3$ | split multiplicative | $4$ | \( 18496 = 2^{6} \cdot 17^{2} \) |
| $17$ | additive | $66$ | \( 192 = 2^{6} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 55488bs
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 1734j2, its twist by $8$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-102}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.6936.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1154594304.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.6542701056.3 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.19628103168.1 | \(\Z/6\Z\) | 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 |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.2929662425445637138754610118221310722048.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.45562110040530548781911696558577824349290496.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.2520656933596540836498892652544.1 | \(\Z/6\Z\) | 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 | split | ord | ord | ord | ord | add | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 7 | 0 | 0 | 0 | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | - | 1 | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.