Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2+222305182134x+266453215936879959\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z+222305182134xz^2+266453215936879959z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+288107516045637x+12431636921138330690838\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(86299249/169, 1573216498391/2197)$ | $7.9955466533508295478221268336$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 55470 \) | = | $2 \cdot 3 \cdot 5 \cdot 43^{2}$ |
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| Discriminant: | $\Delta$ | = | $-31373938148231860089343967232000000000$ | = | $-1 \cdot 2^{53} \cdot 3^{8} \cdot 5^{9} \cdot 43^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{192203697666261893287480365959}{4963160303408775168000000000} \) | = | $2^{-53} \cdot 3^{-8} \cdot 5^{-9} \cdot 17^{3} \cdot 43^{-1} \cdot 339472807^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $5.8750692062827926748644485983$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.9944691484360114631280273416$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0917543736385389$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $8.584268508447272$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.9955466533508295478221268336$ |
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| Real period: | $\Omega$ | ≈ | $0.0049485425920946213586669556578$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 212 $ = $ 53\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.3880562701714720729936560347 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.388056270 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.004949 \cdot 7.995547 \cdot 212}{1^2} \\ & \approx 8.388056270\end{aligned}$$
Modular invariants
Modular form 55470.2.a.v
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1974551040 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $53$ | $I_{53}$ | split multiplicative | -1 | 1 | 53 | 53 |
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $5$ | $1$ | $I_{9}$ | nonsplit multiplicative | 1 | 1 | 9 | 9 |
| $43$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1720 = 2^{3} \cdot 5 \cdot 43 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1377 & 2 \\ 1377 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 1719 & 0 \end{array}\right),\left(\begin{array}{rr} 861 & 2 \\ 861 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1719 & 2 \\ 1718 & 3 \end{array}\right),\left(\begin{array}{rr} 1121 & 2 \\ 1121 & 3 \end{array}\right),\left(\begin{array}{rr} 431 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1720])$ is a degree-$1230331576320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1720\Z)$.
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 | $4$ | \( 9245 = 5 \cdot 43^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 3698 = 2 \cdot 43^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 11094 = 2 \cdot 3 \cdot 43^{2} \) |
| $43$ | additive | $968$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
| $53$ | good | $2$ | \( 27735 = 3 \cdot 5 \cdot 43^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 55470.v consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 1290.f1, its twist by $-43$.
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 |
|---|---|---|---|
| $3$ | 3.1.1720.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.5088448000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | nonsplit | nonsplit | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord | add | ord |
| $\lambda$-invariant(s) | 4 | 3 | 1 | 1 | 1 | 5 | 1,1 | 1 | 3 | 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.