Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2+94x+121\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z+94xz^2+121z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+1509x+9270\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(7, 29)$ | $0.46344372956874925538465392637$ | $\infty$ |
| $(-5/4, 1/8)$ | $0$ | $2$ |
Integral points
\( \left(-1, 5\right) \), \( \left(-1, -5\right) \), \( \left(7, 29\right) \), \( \left(7, -37\right) \), \( \left(11, 43\right) \), \( \left(11, -55\right) \), \( \left(403, 7883\right) \), \( \left(403, -8287\right) \)
Invariants
| Conductor: | $N$ | = | \( 1386 \) | = | $2 \cdot 3^{2} \cdot 7 \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $-62752536$ | = | $-1 \cdot 2^{3} \cdot 3^{3} \cdot 7^{4} \cdot 11^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{3436115229}{2324168} \) | = | $2^{-3} \cdot 3^{3} \cdot 7^{-4} \cdot 11^{-2} \cdot 503^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.18490842802140822881038851339$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.089744644145619194038422795841$ |
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| $abc$ quality: | $Q$ | ≈ | $1.011689787169185$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.490852327061903$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.46344372956874925538465392637$ |
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| Real period: | $\Omega$ | ≈ | $1.2380099121081863145953637655$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 3\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.4424875854629839963991659738 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.442487585 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.238010 \cdot 0.463444 \cdot 24}{2^2} \\ & \approx 3.442487585\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 384 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $11$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 264 = 2^{3} \cdot 3 \cdot 11 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 145 & 4 \\ 26 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 92 & 1 \\ 175 & 0 \end{array}\right),\left(\begin{array}{rr} 261 & 4 \\ 260 & 5 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 131 & 0 \end{array}\right),\left(\begin{array}{rr} 233 & 34 \\ 32 & 231 \end{array}\right)$.
The torsion field $K:=\Q(E[264])$ is a degree-$81100800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/264\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 3 \) |
| $3$ | additive | $6$ | \( 77 = 7 \cdot 11 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 198 = 2 \cdot 3^{2} \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 1386f
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1386a2, its twist by $-3$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.104544.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.369975361536.15 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.699484667904.4 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.76879700667.2 | \(\Z/6\Z\) | 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 | split | add | ord | nonsplit | split | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | - | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 1 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.