Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-803355x+277146058\)
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(homogenize, simplify) |
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\(y^2z=x^3-803355xz^2+277146058z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-803355x+277146058\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(518, 0)$ | $0$ | $2$ |
Integral points
\( \left(518, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 77616 \) | = | $2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $100757479882752$ | = | $2^{18} \cdot 3^{3} \cdot 7^{6} \cdot 11^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{4406910829875}{7744} \) | = | $2^{-6} \cdot 3^{3} \cdot 5^{3} \cdot 11^{-2} \cdot 1093^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9461612909758895772064415399$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.0054059637212601923877217374896$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0492563620983195$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.654125372177722$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.51165008073223266889953555833$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.0466003229289306755981422333 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.046600323 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.511650 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 2.046600323\end{aligned}$$
Modular invariants
Modular form 77616.2.a.ea
For more coefficients, see the Downloads section to the right.
| Modular degree: | 552960 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{10}^{*}$ | additive | -1 | 4 | 18 | 6 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $2$ | $I_{2}$ | 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 \( 924 = 2^{2} \cdot 3 \cdot 7 \cdot 11 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 846 & 791 \\ 847 & 538 \end{array}\right),\left(\begin{array}{rr} 673 & 672 \\ 210 & 337 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 874 & 915 \end{array}\right),\left(\begin{array}{rr} 913 & 12 \\ 912 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 263 & 0 \\ 0 & 923 \end{array}\right),\left(\begin{array}{rr} 848 & 539 \\ 273 & 232 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[924])$ is a degree-$1277337600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/924\Z)$.
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 | $2$ | \( 147 = 3 \cdot 7^{2} \) |
| $3$ | additive | $6$ | \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \) |
| $7$ | additive | $26$ | \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 77616dq
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 198d4, its twist by $-84$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.4.640332.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{7})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.702900120384.4 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.867491057664.36 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.6560401123584.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.6560401123584.1 | \(\Z/12\Z\) | 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 |
| $18$ | 18.6.29880054062425960412864218298610352128.1 | \(\Z/18\Z\) | 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 | 7 | 11 |
|---|---|---|---|---|
| Reduction type | add | add | add | split |
| $\lambda$-invariant(s) | - | - | - | 1 |
| $\mu$-invariant(s) | - | - | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.