Base field \(\Q(\sqrt{-3}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 1 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(6 a : 26 a - 13 : 1\right)$ | $0.10756997376092766753586559536947339257$ | $\infty$ |
| $\left(-26 a : 26 a - 77 : 1\right)$ | $1.1887004589488550673030416174013041568$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((338)\) | = | \((2)\cdot(-4a+1)^{2}\cdot(4a-3)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 114244 \) | = | \(4\cdot13^{2}\cdot13^{2}\) |
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| Discriminant: | $\Delta$ | = | $-2768896$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-2768896)\) | = | \((2)^{14}\cdot(-4a+1)^{2}\cdot(4a-3)^{2}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 7666785058816 \) | = | \(4^{14}\cdot13^{2}\cdot13^{2}\) |
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| j-invariant: | $j$ | = | \( -\frac{38575685889}{16384} \) | ||
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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: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 2 \) |
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| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.12786847717873101559746660743636054477 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.511473908714924062389866429745442179080 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 1.34890460723721127332472205534401207062 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 14 \) = \(( 2 \cdot 7 )\cdot1\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 5.5766338522240486165796698414229802079 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}5.576633852 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 1.348905 \cdot 0.511474 \cdot 14 } { {1^2 \cdot 1.732051} } \\ & \approx 5.576633852 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(4\) | \(14\) | \(I_{14}\) | Split multiplicative | \(-1\) | \(1\) | \(14\) | \(14\) |
| \((-4a+1)\) | \(13\) | \(1\) | \(II\) | Additive | \(1\) | \(2\) | \(2\) | \(0\) |
| \((4a-3)\) | \(13\) | \(1\) | \(II\) | Additive | \(1\) | \(2\) | \(2\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(7\) | 7B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
114244.3-f
consists of curves linked by isogenies of
degree 7.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 338.c1 |
| \(\Q\) | 3042.k1 |