Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-5448x-113258\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-5448xz^2-113258z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7059987x-5262973650\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-34, 198)$ | $0.30287473363198196795871054742$ | $\infty$ |
| $(-23, 11)$ | $0$ | $2$ |
Integral points
\( \left(-55, 171\right) \), \( \left(-55, -117\right) \), \( \left(-34, 198\right) \), \( \left(-34, -165\right) \), \( \left(-23, 11\right) \), \( \left(98, 495\right) \), \( \left(98, -594\right) \), \( \left(266, 4023\right) \), \( \left(266, -4290\right) \), \( \left(329, 5643\right) \), \( \left(329, -5973\right) \)
Invariants
| Conductor: | $N$ | = | \( 726 \) | = | $2 \cdot 3 \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $4849031734272$ | = | $2^{10} \cdot 3^{5} \cdot 11^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{10091699281}{2737152} \) | = | $2^{-10} \cdot 3^{-5} \cdot 11^{-1} \cdot 2161^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1420819465786827428458677765$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.056865689820502529185104012482$ |
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| $abc$ quality: | $Q$ | ≈ | $1.073399617726021$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.680769120855953$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.30287473363198196795871054742$ |
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| Real period: | $\Omega$ | ≈ | $0.56700993613783485390447295831$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 40 $ = $ 2\cdot5\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.7173298337443383786694291511 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.717329834 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.567010 \cdot 0.302875 \cdot 40}{2^2} \\ & \approx 1.717329834\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2400 |
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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 not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $3$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $11$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 |
|---|---|---|
| $2$ | 2B | 8.6.0.4 |
| $5$ | 5B.4.1 | 5.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 1301 & 20 \\ 1300 & 21 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 896 & 5 \\ 1275 & 1306 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 584 & 1315 \\ 1245 & 14 \end{array}\right),\left(\begin{array}{rr} 531 & 20 \\ 1300 & 1187 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 1080 & 971 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 661 & 20 \\ 10 & 201 \end{array}\right),\left(\begin{array}{rr} 991 & 20 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1320])$ is a degree-$1622016000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1320\Z)$.
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 | $4$ | \( 363 = 3 \cdot 11^{2} \) |
| $3$ | split multiplicative | $4$ | \( 242 = 2 \cdot 11^{2} \) |
| $5$ | good | $2$ | \( 121 = 11^{2} \) |
| $11$ | additive | $72$ | \( 6 = 2 \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 726.d
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 66.c3, its twist by $-11$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.2.33.1-132.1-j3 |
| $2$ | \(\Q(\sqrt{-11}) \) | \(\Z/10\Z\) | 2.0.11.1-396.2-b3 |
| $4$ | 4.0.2112.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-11})\) | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $8$ | 8.4.1322463200256.7 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.4857532416.6 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.5021227463472.3 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.539725824.3 | \(\Z/20\Z\) | 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 |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $16$ | 16.0.23595621172490797056.3 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/30\Z\) | not in database |
| $20$ | 20.4.169675210983039290802001953125.1 | \(\Z/10\Z\) | 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 | split | ord | ord | add | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 1 | 4 | 5 | 1 | - | 1 | 1 | 1,1 | 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 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.