Base field \(\Q(\sqrt{-166}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 166 \); class number \(10\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((7,a+3)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{174}{289} a + \frac{111}{289} : \frac{10476}{4913} a + \frac{174530}{4913} : 1\right)$ | $3.7616542980915461577139672446607776515$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((70,a+38)\) | = | \((2,a)\cdot(5,a+3)\cdot(7,a+3)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 70 \) | = | \(2\cdot5\cdot7\) |
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| Discriminant: | $\Delta$ | = | $-1944864a-19085312$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((-1944864a-19085312)\) | = | \((2,a)^{11}\cdot(5,a+3)\cdot(7,a+3)^{13}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 992143466567680 \) | = | \(2^{11}\cdot5\cdot7^{13}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((2240,32a+1216)\) | = | \((2,a)^{11}\cdot(5,a+3)\cdot(7,a+3)\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 71680 \) | = | \(2^{11}\cdot5\cdot7\) |
| j-invariant: | $j$ | = | \( -\frac{60127}{2240} a + \frac{5094}{35} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 3.7616542980915461577139672446607776515 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 7.5233085961830923154279344893215553030 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 21.855706987001724343216236777672679172 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 1 \) = \(1\cdot1\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.1905069911856183252423173216644409245 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}3.190506991 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 10.927853 \cdot 7.523309 \cdot 1 } { {1^2 \cdot 25.768197} } \\ & \approx 3.190506991 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 3 primes $\frak{p}$ of bad reduction. Primes of good reduction for the curve but which divide the discriminant of the model above (if any) are included.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(1\) | \(I_{11}\) | Non-split multiplicative | \(1\) | \(1\) | \(11\) | \(11\) |
| \((5,a+3)\) | \(5\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((7,a+3)\) | \(7\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(11\) | 11B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
11.
Its isogeny class
70.3-a
consists of curves linked by isogenies of
degree 11.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.