Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-129168x+17868033\)
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(homogenize, simplify) |
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\(y^2z=x^3-129168xz^2+17868033z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-129168x+17868033\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(184, 575)$ | $0.68867984308796783354042740658$ | $\infty$ |
| $(207, 0)$ | $0$ | $2$ |
Integral points
\((-414,\pm 621)\), \((184,\pm 575)\), \( \left(207, 0\right) \), \((208,\pm 1)\), \((234,\pm 675)\), \((96928,\pm 30176641)\)
Invariants
| Conductor: | $N$ | = | \( 95220 \) | = | $2^{2} \cdot 3^{2} \cdot 5 \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $2394830610000$ | = | $2^{4} \cdot 3^{9} \cdot 5^{4} \cdot 23^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{62200479744}{625} \) | = | $2^{20} \cdot 3^{3} \cdot 5^{-4} \cdot 13^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5344720262394582168594742731$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.30440980443055991086105856970$ |
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| $abc$ quality: | $Q$ | ≈ | $1.239581518021865$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.092850180315831$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.68867984308796783354042740658$ |
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| Real period: | $\Omega$ | ≈ | $0.73811399755000008689511310290$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 3\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.0499453920826004025795041965 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.049945392 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.738114 \cdot 0.688680 \cdot 24}{2^2} \\ & \approx 3.049945392\end{aligned}$$
Modular invariants
Modular form 95220.2.a.a
For more coefficients, see the Downloads section to the right.
| Modular degree: | 359424 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $23$ | $2$ | $III$ | 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 | 8.12.0.34 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \), index $96$, genus $5$, and generators
$\left(\begin{array}{rr} 2656 & 5 \\ 531 & 16 \end{array}\right),\left(\begin{array}{rr} 4417 & 16 \\ 2216 & 129 \end{array}\right),\left(\begin{array}{rr} 1850 & 9 \\ 1811 & 5494 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 5396 & 5465 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4142 \\ 2 & 2075 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 8 & 1509 \end{array}\right),\left(\begin{array}{rr} 13 & 4 \\ 5380 & 5477 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 5456 & 5427 \end{array}\right),\left(\begin{array}{rr} 5505 & 16 \\ 5504 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[5520])$ is a degree-$1575820984320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5520\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 69 = 3 \cdot 23 \) |
| $3$ | additive | $2$ | \( 10580 = 2^{2} \cdot 5 \cdot 23^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 19044 = 2^{2} \cdot 3^{2} \cdot 23^{2} \) |
| $23$ | additive | $156$ | \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 95220.a
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{69}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.21024576.4 | \(\Z/4\Z\) | not in database |
| $8$ | 8.4.442032795979776.14 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.442032795979776.23 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 |
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 | add | add | nonsplit | ord | ord | ord | ord | ord | add | ss | ord | ss | ss | ord | ord |
| $\lambda$-invariant(s) | - | - | 1 | 1 | 1 | 3 | 1 | 1 | - | 3,1 | 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.