Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2+68863x-46738593\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z+68863xz^2-46738593z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+5577876x-34089167952\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(297, 0)$ | $0$ | $2$ |
Integral points
\( \left(297, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 35904 \) | = | $2^{6} \cdot 3 \cdot 11 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-965528762147930112$ | = | $-1 \cdot 2^{22} \cdot 3 \cdot 11 \cdot 17^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{137763859017023}{3683199928848} \) | = | $2^{-4} \cdot 3^{-1} \cdot 11^{-1} \cdot 17^{-8} \cdot 51647^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1310768652535971124955304837$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0913560944136791483696823015$ |
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| $abc$ quality: | $Q$ | ≈ | $1.009169658758712$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.656891341896092$ | |||
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: | $\Omega$ | ≈ | $0.13466373972436487885895181582$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot1\cdot1\cdot2^{3} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $4.3092396711796761234864581061 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 4.309239671 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.134664 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 4.309239671\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 393216 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{12}^{*}$ | additive | 1 | 6 | 22 | 4 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $17$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
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 | 8.48.0.221 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8976 = 2^{4} \cdot 3 \cdot 11 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 8961 & 16 \\ 8960 & 17 \end{array}\right),\left(\begin{array}{rr} 8975 & 8960 \\ 6724 & 6603 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5618 & 6739 \\ 3391 & 4510 \end{array}\right),\left(\begin{array}{rr} 1057 & 16 \\ 8456 & 129 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 3272 & 1 \\ 895 & 10 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 8972 & 8973 \end{array}\right),\left(\begin{array}{rr} 5992 & 1 \\ 3071 & 10 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 8878 & 8963 \end{array}\right)$.
The torsion field $K:=\Q(E[8976])$ is a degree-$6353112268800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8976\Z)$.
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 | $2$ | \( 33 = 3 \cdot 11 \) |
| $3$ | split multiplicative | $4$ | \( 11968 = 2^{6} \cdot 11 \cdot 17 \) |
| $11$ | split multiplicative | $12$ | \( 3264 = 2^{6} \cdot 3 \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 2112 = 2^{6} \cdot 3 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 35904.cz
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 1122.e6, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-33}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-66}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-33})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{33})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\zeta_{8})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.84637644816384.65 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.21159411204096.47 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.77720518656.8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | 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 | 11 | 17 |
|---|---|---|---|---|
| Reduction type | add | split | split | split |
| $\lambda$-invariant(s) | - | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ 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 $0$.