Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+117x-918\)
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(homogenize, simplify) |
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\(y^2z=x^3+117xz^2-918z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+117x-918\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(6, 0)$ | $0$ | $2$ |
Integral points
\( \left(6, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 720 \) | = | $2^{4} \cdot 3^{2} \cdot 5$ |
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| Discriminant: | $\Delta$ | = | $-466560000$ | = | $-1 \cdot 2^{10} \cdot 3^{6} \cdot 5^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{237276}{625} \) | = | $2^{2} \cdot 3^{3} \cdot 5^{-4} \cdot 13^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.34709236233786493295914268225$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.77983643246281100391950670409$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0467057018818753$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.129275421077717$ | |||
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: | $\Omega$ | ≈ | $0.85702594111885301932764735221$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.7140518822377060386552947044 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.714051882 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.857026 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 1.714051882\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 256 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $2$ | $I_{4}$ | nonsplit 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 | 4.24.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 240 = 2^{4} \cdot 3 \cdot 5 \), index $192$, genus $3$, and generators
$\left(\begin{array}{rr} 97 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 16 \\ 64 & 205 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 4 & 49 \end{array}\right),\left(\begin{array}{rr} 225 & 16 \\ 224 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 79 & 0 \\ 0 & 239 \end{array}\right),\left(\begin{array}{rr} 104 & 207 \\ 15 & 224 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 73 & 144 \\ 12 & 13 \end{array}\right)$.
The torsion field $K:=\Q(E[240])$ is a degree-$2949120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/240\Z)$.
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 | $2$ | \( 9 = 3^{2} \) |
| $3$ | additive | $6$ | \( 80 = 2^{4} \cdot 5 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 144 = 2^{4} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 720d
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 40a4, its twist by $12$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.0.4.1-16200.2-a3 |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/4\Z\) | 2.2.12.1-200.1-a1 |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/4\Z\) | 2.0.3.1-6400.1-g1 |
| $4$ | \(\Q(\zeta_{12})\) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.3317760000.2 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.207360000.4 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.5598720000.2 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.0.11007531417600000000.8 | \(\Z/4\Z \oplus \Z/8\Z\) | 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 |
|---|---|---|---|
| Reduction type | add | add | nonsplit |
| $\lambda$-invariant(s) | - | - | 0 |
| $\mu$-invariant(s) | - | - | 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 $0$.