Base field 4.4.9301.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} + x + 3 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{20795407}{4558225} a^{3} + \frac{57398801}{4558225} a^{2} + \frac{10103699}{651175} a + \frac{1142887}{4558225} : -\frac{84743198868}{9731810375} a^{3} - \frac{651428103351}{9731810375} a^{2} + \frac{74426791426}{1390258625} a + \frac{770746792838}{9731810375} : 1\right)$ | $4.7659824764183009039773036489004156655$ | $\infty$ |
| $\left(4 a^{2} - 2 a - 6 : -2 a^{3} - a^{2} + 4 a + 3 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((a^2+a-3)\) | = | \((-a)\cdot(a^3-a^2-4a+2)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 51 \) | = | \(3\cdot17\) |
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| Discriminant: | $\Delta$ | = | $-6a^3+9a^2+28a-42$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-6a^3+9a^2+28a-42)\) | = | \((-a)^{4}\cdot(a^3-a^2-4a+2)^{3}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 397953 \) | = | \(3^{4}\cdot17^{3}\) |
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| j-invariant: | $j$ | = | \( \frac{3145980571677213592387}{397953} a^{3} - \frac{5613926945615477005246}{397953} a^{2} - \frac{11325912207298148829797}{397953} a + \frac{12030886882774010107801}{397953} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 4.7659824764183009039773036489004156655 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 19.063929905673203615909214595601662662 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 2.3143612785362043803110473302721507668 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(2^{2}\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 4.11738306883644 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 9 \) (rounded) |
BSD formula
$$\begin{aligned}4.117383069 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 9 \cdot 2.314361 \cdot 19.063930 \cdot 4 } { {2^2 \cdot 96.441692} } \\ & \approx 4.117383069 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-a)\) | \(3\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((a^3-a^2-4a+2)\) | \(17\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
51.1-c
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.