Base field \(\Q(\sqrt{-1}) \)
Generator \(i\), with minimal polynomial \( x^{2} + 1 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-6 : 3 i - 14 : 1\right)$ | $0.74629920869451370890943760363155317689$ | $\infty$ |
| $\left(\frac{9}{2} : -\frac{9}{4} i - \frac{49}{4} : 1\right)$ | $0$ | $4$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((21)\) | = | \((3)\cdot(7)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 441 \) | = | \(9\cdot49\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||||
| Discriminant: | $\Delta$ | = | $-17294403$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-17294403)\) | = | \((3)\cdot(7)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||||
| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 299096375126409 \) | = | \(9\cdot49^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||||
| j-invariant: | $j$ | = | \( -\frac{4354703137}{17294403} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||||
| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | ||
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 1 \) |
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | $r$ | = | \(1\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.74629920869451370890943760363155317689 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 1.49259841738902741781887520726310635378 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 1.72415385927425396127231453418303639638 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 8 \) = \(1\cdot2^{3}\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 0.64336733042198383881173683541754212198 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$\displaystyle 0.643367330 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 1.724154 \cdot 1.492598 \cdot 8 } { {4^2 \cdot 2.000000} } \approx 0.643367330$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((3)\) | \(9\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((7)\) | \(49\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
441.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 21.a4 |
| \(\Q\) | 336.a4 |