Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-14615x+672573\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-14615xz^2+672573z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-18941715x+31663688238\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(31, 485)$ | $1.0548782518451302868861509182$ | $\infty$ |
| $(267/4, -267/8)$ | $0$ | $2$ |
Integral points
\( \left(31, 485\right) \), \( \left(31, -516\right) \), \( \left(97, 375\right) \), \( \left(97, -472\right) \)
Invariants
| Conductor: | $N$ | = | \( 11154 \) | = | $2 \cdot 3 \cdot 11 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $840697815672$ | = | $2^{3} \cdot 3^{3} \cdot 11^{6} \cdot 13^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{157158018407125}{382657176} \) | = | $2^{-3} \cdot 3^{-3} \cdot 5^{3} \cdot 11^{-6} \cdot 43^{3} \cdot 251^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1666396153106817629351714660$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.52540227594529757892179960561$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9993281682598464$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.333160506102324$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0548782518451302868861509182$ |
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| Real period: | $\Omega$ | ≈ | $0.89324413112330512942163957144$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 1\cdot1\cdot( 2 \cdot 3 )\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.8267914225308233471777117948 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.826791423 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.893244 \cdot 1.054878 \cdot 12}{2^2} \\ & \approx 2.826791423\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 17280 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $11$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $13$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
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 |
| $3$ | 3Ns | 3.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \), index $144$, genus $5$, and generators
$\left(\begin{array}{rr} 7 & 12 \\ 144 & 247 \end{array}\right),\left(\begin{array}{rr} 804 & 5 \\ 2995 & 3360 \end{array}\right),\left(\begin{array}{rr} 5 & 12 \\ 3384 & 3317 \end{array}\right),\left(\begin{array}{rr} 6 & 577 \\ 1759 & 36 \end{array}\right),\left(\begin{array}{rr} 937 & 12 \\ 2190 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 3421 & 12 \\ 3420 & 13 \end{array}\right),\left(\begin{array}{rr} 10 & 3 \\ 1689 & 3424 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1574 & 3015 \\ 3003 & 3290 \end{array}\right)$.
The torsion field $K:=\Q(E[3432])$ is a degree-$177124147200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3432\Z)$.
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 | $4$ | \( 39 = 3 \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 169 = 13^{2} \) |
| $11$ | split multiplicative | $12$ | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
| $13$ | additive | $50$ | \( 66 = 2 \cdot 3 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 11154l
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{78}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.25520352.2 | \(\Z/4\Z\) | not in database |
| $4$ | 4.0.19773.1 | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.6591.1 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.390971529.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.1601419382784.39 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.4.1601419382784.23 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | ss | ss | split | add | ss | ss | ss | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 1 | 1,1 | 3,1 | 2 | - | 1,1 | 1,1 | 5,1 | 1 | 1,1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0,0 | 0,0 | 0 | - | 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.