Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-2108929x-972923809\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-2108929xz^2-972923809z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-170823276x-708748986960\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-958, 13005)$ | $2.6298438387501564940275334448$ | $\infty$ |
| $(2221, 72828)$ | $4.4891722546111744604510446971$ | $\infty$ |
| $(-533, 0)$ | $0$ | $2$ |
| $(1643, 0)$ | $0$ | $2$ |
Integral points
\( \left(-1111, 0\right) \), \((-958,\pm 13005)\), \((-661,\pm 11520)\), \((-535,\pm 1584)\), \( \left(-533, 0\right) \), \( \left(1643, 0\right) \), \((2221,\pm 72828)\), \((9293,\pm 884340)\), \((12011,\pm 1306368)\), \((628331,\pm 498060288)\)
Invariants
| Conductor: | $N$ | = | \( 55488 \) | = | $2^{6} \cdot 3 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $191964672983411195904$ | = | $2^{22} \cdot 3^{8} \cdot 17^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{163936758817}{30338064} \) | = | $2^{-4} \cdot 3^{-8} \cdot 13^{3} \cdot 17^{-2} \cdot 421^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6102585479495529490712104263$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.15393110508152694482059493518$ |
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| $abc$ quality: | $Q$ | ≈ | $1.07570813760169$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.062163166618042$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $10.320803029838108926574058123$ |
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| Real period: | $\Omega$ | ≈ | $0.12688133554167618264215540887$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2^{3}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $10.476138178307498251867455055 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.476138178 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.126881 \cdot 10.320803 \cdot 128}{4^2} \\ & \approx 10.476138178\end{aligned}$$
Modular invariants
Modular form 55488.2.a.cr
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1769472 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{12}^{*}$ | additive | 1 | 6 | 22 | 4 |
| $3$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $17$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 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 |
|---|---|---|
| $2$ | 2Cs | 8.96.0.104 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 136 = 2^{3} \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5 & 108 \\ 130 & 27 \end{array}\right),\left(\begin{array}{rr} 63 & 96 \\ 128 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 81 & 134 \\ 122 & 5 \end{array}\right),\left(\begin{array}{rr} 129 & 8 \\ 128 & 9 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 132 & 133 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[136])$ is a degree-$626688$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/136\Z)$.
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 | $2$ | \( 289 = 17^{2} \) |
| $3$ | split multiplicative | $4$ | \( 18496 = 2^{6} \cdot 17^{2} \) |
| $17$ | additive | $162$ | \( 192 = 2^{6} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 55488bl
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 102b2, its twist by $136$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{34}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-17})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-17})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.5473632256.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | ss | ord | ord | add | ord | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 3 | 2 | 2,2 | 2 | 2 | - | 2 | 2,2 | 2 | 2 | 2 | 2 | 2 | 2,2 |
| $\mu$-invariant(s) | - | 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.