Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+2003397x-1268243350\)
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(homogenize, simplify) |
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\(y^2z=x^3+2003397xz^2-1268243350z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+2003397x-1268243350\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(550, 0)$ | $0$ | $2$ |
Integral points
\( \left(550, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 17424 \) | = | $2^{4} \cdot 3^{2} \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $-1209459925880563433472$ | = | $-1 \cdot 2^{17} \cdot 3^{16} \cdot 11^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{168105213359}{228637728} \) | = | $2^{-5} \cdot 3^{-10} \cdot 11^{-2} \cdot 5519^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7311088617526555526693385771$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.28970790045947012552351204820$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1002106061618209$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.67757443354806$ | |||
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: | $\Omega$ | ≈ | $0.081840834815592866650935930945$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.6189067140989717328299497902 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.618906714 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.081841 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 2.618906714\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 921600 |
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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 | 4 | 17 | 5 |
| $3$ | $4$ | $I_{10}^{*}$ | additive | -1 | 2 | 16 | 10 |
| $11$ | $4$ | $I_{2}^{*}$ | 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 |
|---|---|---|
| $2$ | 2B | 8.6.0.5 |
| $5$ | 5B.4.1 | 5.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 1301 & 20 \\ 1300 & 21 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 1080 & 971 \end{array}\right),\left(\begin{array}{rr} 654 & 1315 \\ 935 & 1274 \end{array}\right),\left(\begin{array}{rr} 16 & 5 \\ 615 & 1306 \end{array}\right),\left(\begin{array}{rr} 119 & 1300 \\ 1190 & 1119 \end{array}\right),\left(\begin{array}{rr} 531 & 20 \\ 1300 & 1187 \end{array}\right),\left(\begin{array}{rr} 439 & 1300 \\ 430 & 1119 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1320])$ is a degree-$1622016000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1320\Z)$.
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 | $4$ | \( 1089 = 3^{2} \cdot 11^{2} \) |
| $3$ | additive | $8$ | \( 1936 = 2^{4} \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 144 = 2^{4} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 17424cf
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 66c2, its twist by $-132$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-33}) \) | \(\Z/10\Z\) | not in database |
| $4$ | 4.2.8712.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-33})\) | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $8$ | 8.0.19896452775936.50 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.4857532416.8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.4857532416.12 | \(\Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/30\Z\) | not in database |
| $20$ | 20.4.10505841838220888855090208000000000000000.1 | \(\Z/10\Z\) | 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 | 11 |
|---|---|---|---|---|
| Reduction type | add | add | ord | add |
| $\lambda$-invariant(s) | - | - | 0 | - |
| $\mu$-invariant(s) | - | - | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.
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 $0$.