Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2+3x+9\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z+3xz^2+9z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+3213x+368334\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(0, 3\right) \) | $0.28467632522426411770903856556$ | $\infty$ |
| \( \left(-2, 1\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([0:3:1]\) | $0.28467632522426411770903856556$ | $\infty$ |
| \([-2:1:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(15, 648\right) \) | $0.28467632522426411770903856556$ | $\infty$ |
| \( \left(-57, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-2, 1\right) \), \( \left(-1, 3\right) \), \( \left(-1, -2\right) \), \( \left(0, 3\right) \), \( \left(0, -3\right) \), \( \left(3, 6\right) \), \( \left(3, -9\right) \), \( \left(8, 21\right) \), \( \left(8, -29\right) \), \( \left(30, 153\right) \), \( \left(30, -183\right) \)
\([-2:1:1]\), \([-1:3:1]\), \([-1:-2:1]\), \([0:3:1]\), \([0:-3:1]\), \([3:6:1]\), \([3:-9:1]\), \([8:21:1]\), \([8:-29:1]\), \([30:153:1]\), \([30:-183:1]\)
\( \left(-57, 0\right) \), \((-21,\pm 540)\), \((15,\pm 648)\), \((123,\pm 1620)\), \((303,\pm 5400)\), \((1095,\pm 36288)\)
Invariants
| Conductor: | $N$ | = | \( 930 \) | = | $2 \cdot 3 \cdot 5 \cdot 31$ |
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| Minimal Discriminant: | $\Delta$ | = | $-27900$ | = | $-1 \cdot 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 31 $ |
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| j-invariant: | $j$ | = | \( \frac{1685159}{27900} \) | = | $2^{-2} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{3} \cdot 17^{3} \cdot 31^{-1}$ |
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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: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.46642710039823872362700230669$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.46642710039823872362700230669$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8654261723900244$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.5830352513217343$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.28467632522426411770903856556$ |
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| Real period: | $\Omega$ | ≈ | $2.7841974935628569258308688264$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.5851902223321617211358399972 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.585190222 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.784197 \cdot 0.284676 \cdot 8}{2^2} \\ & \approx 1.585190222\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 96 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $31$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1240 = 2^{3} \cdot 5 \cdot 31 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1122 & 1 \\ 679 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 497 & 4 \\ 994 & 9 \end{array}\right),\left(\begin{array}{rr} 621 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 777 & 466 \\ 464 & 775 \end{array}\right),\left(\begin{array}{rr} 1237 & 4 \\ 1236 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[1240])$ is a degree-$54853632000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1240\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 31 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 310 = 2 \cdot 5 \cdot 31 \) |
| $5$ | split multiplicative | $6$ | \( 186 = 2 \cdot 3 \cdot 31 \) |
| $31$ | split multiplicative | $32$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 930.c
consists of 2 curves linked by isogenies of
degree 2.
Twists
This elliptic curve is its own minimal quadratic twist.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-31}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.49600.2 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.2364213760000.5 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.1635989745870000.10 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | nonsplit | split | ord | ord | ord | ss | ss | ord | ord | split | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 8 | 1 | 2 | 1 | 3 | 1 | 1,1 | 3,1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.