Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-78833x+8545662\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-78833xz^2+8545662z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6385500x+6210631125\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-94, 3888)$ | $4.0553577884113215131329661118$ | $\infty$ |
| $(162, 0)$ | $0$ | $2$ |
Integral points
\((-94,\pm 3888)\), \( \left(162, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 66300 \) | = | $2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $684713250000$ | = | $2^{4} \cdot 3^{6} \cdot 5^{6} \cdot 13 \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{216727177216000}{2738853} \) | = | $2^{14} \cdot 3^{-6} \cdot 5^{3} \cdot 11^{3} \cdot 13^{-1} \cdot 17^{-2} \cdot 43^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4172911968647406534732565752$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.38152318046104202970046620143$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9818588720143953$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.092875249013007$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.0553577884113215131329661118$ |
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| Real period: | $\Omega$ | ≈ | $0.82469637925044331981165051715$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot2\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.6888777693358045640786448822 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.688877769 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.824696 \cdot 4.055358 \cdot 8}{2^2} \\ & \approx 6.688877769\end{aligned}$$
Modular invariants
Modular form 66300.2.a.t
For more coefficients, see the Downloads section to the right.
| Modular degree: | 207360 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $13$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $17$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 13260 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \cdot 17 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 11501 & 10610 \\ 11550 & 10621 \end{array}\right),\left(\begin{array}{rr} 1551 & 10390 \\ 10850 & 1121 \end{array}\right),\left(\begin{array}{rr} 5303 & 0 \\ 0 & 13259 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 12481 & 10620 \\ 630 & 10681 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 13210 & 13251 \end{array}\right),\left(\begin{array}{rr} 826 & 2655 \\ 4665 & 5296 \end{array}\right),\left(\begin{array}{rr} 13249 & 12 \\ 13248 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[13260])$ is a degree-$47301808619520$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/13260\Z)$.
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 | $2$ | \( 325 = 5^{2} \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 22100 = 2^{2} \cdot 5^{2} \cdot 13 \cdot 17 \) |
| $5$ | additive | $14$ | \( 2652 = 2^{2} \cdot 3 \cdot 13 \cdot 17 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 5100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 66300f
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 2652f1, its twist by $5$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.6011200.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{13})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.128813937174000.2 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.6106734799360000.200 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.36134525440000.99 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | 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 |
| $18$ | 18.6.9372723775173815911068795516109512000000000.1 | \(\Z/18\Z\) | 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 | add | nonsplit | add | ord | ss | nonsplit | split | ord | ss | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | - | 1 | - | 1 | 1,1 | 1 | 2 | 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 | 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.