Base field 4.4.6125.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 9 x^{2} + 9 x + 11 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-5 a^{3} - 7 a^{2} + 32 a + 31 : -13 a^{3} - 17 a^{2} + 82 a + 73 : 1\right)$ | $0.36737780216497338453285000149795958011$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((a^3+a^2-8a-3)\) | = | \((3a^3+4a^2-17a-13)\cdot(-a^3-2a^2+5a+12)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 245 \) | = | \(5\cdot49\) |
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| Discriminant: | $\Delta$ | = | $-35$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-35)\) | = | \((3a^3+4a^2-17a-13)^{4}\cdot(-a^3-2a^2+5a+12)^{2}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 1500625 \) | = | \(5^{4}\cdot49^{2}\) |
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| j-invariant: | $j$ | = | \( -\frac{262144}{35} \) | ||
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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: | $r_{\mathrm{an}}$ | = | \( 1 \) |
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| Mordell-Weil rank: | $r$ | = | \(1\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.36737780216497338453285000149795958011 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 1.46951120865989353813140000599183832044 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 23.641185852562422186317149524468730185 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 8 \) = \(2^{2}\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.55123245124208 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}3.551232451 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 23.641186 \cdot 1.469511 \cdot 8 } { {1^2 \cdot 78.262379} } \\ & \approx 3.551232451 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 2 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((3a^3+4a^2-17a-13)\) | \(5\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((-a^3-2a^2+5a+12)\) | \(49\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
245.1-b
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 4 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 245.c2 |
| \(\Q\) | 1225.e2 |
| \(\Q(\sqrt{5}) \) | a curve with conductor norm 12005 (not in the database) |
| \(\Q(\sqrt{5}) \) | 2.2.5.1-1225.1-b2 |