Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+244458x-115728134\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+244458xz^2-115728134z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+3911325x-7402689250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{18561}{49}, \frac{1848349}{343}\right) \) | $4.7648525459114675024236988617$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([129927:1848349:343]\) | $4.7648525459114675024236988617$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{74195}{49}, \frac{15306500}{343}\right) \) | $4.7648525459114675024236988617$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 54450 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-6714624479449218750$ | = | $-1 \cdot 2 \cdot 3^{6} \cdot 5^{9} \cdot 11^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{80062991}{332750} \) | = | $2^{-1} \cdot 5^{-3} \cdot 11^{-3} \cdot 431^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2951734106883607158793169703$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.25779932626192958914965710376$ |
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| $abc$ quality: | $Q$ | ≈ | $0.922044169862911$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.645155490630335$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.7648525459114675024236988617$ |
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| Real period: | $\Omega$ | ≈ | $0.11994701807183392396038184988$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot1\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.5722388354725333950805333985 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.572238835 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.119947 \cdot 4.764853 \cdot 8}{1^2} \\ & \approx 4.572238835\end{aligned}$$
Modular invariants
Modular form 54450.2.a.bn
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1036800 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $2$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $11$ | $4$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
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 |
|---|---|---|---|
| $3$ | 3B | 3.4.0.1 | $4$ |
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 $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 382 & 933 \\ 1145 & 653 \end{array}\right),\left(\begin{array}{rr} 991 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1315 & 6 \\ 1314 & 7 \end{array}\right),\left(\begin{array}{rr} 661 & 6 \\ 663 & 19 \end{array}\right),\left(\begin{array}{rr} 263 & 1314 \\ 789 & 1301 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 119 & 1314 \\ 357 & 1301 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1320])$ is a degree-$29196288000$ 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 | $4$ | \( 27225 = 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
| $3$ | additive | $6$ | \( 6050 = 2 \cdot 5^{2} \cdot 11^{2} \) |
| $5$ | additive | $18$ | \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 54450.bn
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 110.c2, its twist by $165$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-55}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.440.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.85184000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.5821794000.2 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.10648000.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | 12.0.7256313856000000.4 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.82953159205371192601654509210937500000000.2 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.3232886381878746057670656000000000.2 | \(\Z/6\Z\) | 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 | add | add | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | - | - | 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 |
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.