Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-920735x-279119203\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-920735xz^2-279119203z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1193273235x-13004686440018\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{1465231}{900}, \frac{1338860281}{27000}\right) \) | $11.345188953598667443086750625$ | $\infty$ |
| \( \left(-\frac{1397}{4}, \frac{1397}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([43956930:1338860281:27000]\) | $11.345188953598667443086750625$ | $\infty$ |
| \([-2794:1397:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{1465606}{25}, \frac{1360838746}{125}\right) \) | $11.345188953598667443086750625$ | $\infty$ |
| \( \left(-12558, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 51842 \) | = | $2 \cdot 7^{2} \cdot 23^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $16392058045634853512$ | = | $2^{3} \cdot 7^{12} \cdot 23^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{4956477625}{941192} \) | = | $2^{-3} \cdot 5^{3} \cdot 7^{-6} \cdot 11^{3} \cdot 31^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4044970185812611098392125240$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.13620516391097038811684026363$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0082122835525031$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.864830678599058$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $11.345188953598667443086750625$ |
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| Real period: | $\Omega$ | ≈ | $0.15615162138944456479671830098$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.0862786002961915169947611778 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.086278600 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.156152 \cdot 11.345189 \cdot 16}{2^2} \\ & \approx 7.086278600\end{aligned}$$
Modular invariants
Modular form 51842.2.a.j
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1216512 |
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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 3 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 |
| $7$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $23$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.6.0.6 | $6$ |
| $3$ | 3Cs | 3.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 11592 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 23 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11557 & 36 \\ 11556 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 1007 & 0 \\ 0 & 11591 \end{array}\right),\left(\begin{array}{rr} 829 & 276 \\ 4554 & 6211 \end{array}\right),\left(\begin{array}{rr} 5797 & 9108 \\ 0 & 7085 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 19 & 24 \\ 1440 & 1819 \end{array}\right),\left(\begin{array}{rr} 8074 & 6555 \\ 10833 & 1864 \end{array}\right),\left(\begin{array}{rr} 3587 & 2484 \\ 2346 & 11315 \end{array}\right)$.
The torsion field $K:=\Q(E[11592])$ is a degree-$3722877075456$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/11592\Z)$.
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 | $4$ | \( 25921 = 7^{2} \cdot 23^{2} \) |
| $3$ | good | $2$ | \( 25921 = 7^{2} \cdot 23^{2} \) |
| $7$ | additive | $32$ | \( 1058 = 2 \cdot 23^{2} \) |
| $23$ | additive | $266$ | \( 98 = 2 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 51842.j
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 14.a3, its twist by $161$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{161}) \) | \(\Z/6\Z\) | 2.2.161.1-28.1-b4 |
| $2$ | \(\Q(\sqrt{-483}) \) | \(\Z/6\Z\) | 2.0.483.1-28.1-d4 |
| $4$ | \(\Q(\sqrt{-46 +46 \sqrt{-7}})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{161})\) | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{161})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-483})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.4.2818145479819264.34 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.44033523122176.31 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | \(\Q(\sqrt{2}, \sqrt{-3}, \sqrt{161})\) | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.688023798784.4 | \(\Z/12\Z\) | not in database |
| $8$ | 8.0.55729927701504.11 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.3312864776359272743499043310391801.1 | \(\Z/18\Z\) | not in database |
| $18$ | 18.0.267088352842053899920554678183697690963968.2 | \(\Z/18\Z\) | 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 | ord | ss | add | ss | ord | ord | ord | add | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 6 | 5 | 1,1 | - | 1,1 | 1 | 1 | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 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.