Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-343\)
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(homogenize, simplify) |
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\(y^2z=x^3-343z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-343\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(14, 49)$ | $0.52689537248059847502449502904$ | $\infty$ |
| $(7, 0)$ | $0$ | $2$ |
Integral points
\( \left(7, 0\right) \), \((8,\pm 13)\), \((14,\pm 49)\), \((28,\pm 147)\), \((154,\pm 1911)\)
Invariants
| Conductor: | $N$ | = | \( 1764 \) | = | $2^{2} \cdot 3^{2} \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $-50824368$ | = | $-1 \cdot 2^{4} \cdot 3^{3} \cdot 7^{6} $ |
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| j-invariant: | $j$ | = | \( 0 \) | = | $0$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-3})/2]\) (potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.15753977845329459688402919225$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.3211174284280379149898691959$ |
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| $abc$ quality: | $Q$ | ≈ | $$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.370898046830738$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.52689537248059847502449502904$ |
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| Real period: | $\Omega$ | ≈ | $0.91794366225234805850875634239$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 3\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.9019616070319336743899652222 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.901961607 \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 0.917944 \cdot 0.526895 \cdot 24}{2^2} \\ & \approx 2.901961607\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 288 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 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 |
|---|---|---|
| $2$ | 2B | 16.192.9.83 |
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 | $2$ | \( 147 = 3 \cdot 7^{2} \) |
| $3$ | additive | $6$ | \( 196 = 2^{2} \cdot 7^{2} \) |
| $7$ | additive | $26$ | \( 36 = 2^{2} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 1764b
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 36a1, its twist by $-7$.
The minimal sextic twist of this elliptic curve is 27.a4, its sextic twist by $-5488$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/6\Z\) | 2.0.7.1-1296.3-a1 |
| $4$ | 4.2.84672.5 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-7})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.12002256.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.7169347584.11 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.7169347584.3 | \(\Z/12\Z\) | not in database |
| $12$ | 12.0.144054149089536.4 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.4149707998464.1 | \(\Z/2\Z \oplus \Z/14\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | 16.0.51399544780206637056.9 | \(\Z/4\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.34031410500946135240470528.1 | \(\Z/18\Z\) | 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 | add | ss | add | ss | ord | ss | ord | ss | ss | ord | ord | ss | ord | ss |
| $\lambda$-invariant(s) | - | - | 3,1 | - | 3,1 | 1 | 1,1 | 1 | 1,1 | 1,1 | 1 | 1 | 1,1 | 1 | 1,1 |
| $\mu$-invariant(s) | - | - | 0,0 | - | 0,0 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.