Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+11092x+573232\)
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(homogenize, simplify) |
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\(y^2z=x^3+11092xz^2+573232z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+11092x+573232\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(6, 800)$ | $1.2584481583354583883252841080$ | $\infty$ |
| $(-44, 0)$ | $0$ | $2$ |
Integral points
\( \left(-44, 0\right) \), \((6,\pm 800)\), \((37,\pm 1017)\), \((294,\pm 5408)\)
Invariants
| Conductor: | $N$ | = | \( 29120 \) | = | $2^{6} \cdot 5 \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-229292277760000$ | = | $-1 \cdot 2^{18} \cdot 5^{4} \cdot 7^{2} \cdot 13^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{575722725759}{874680625} \) | = | $3^{3} \cdot 5^{-4} \cdot 7^{-2} \cdot 13^{-4} \cdot 47^{3} \cdot 59^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4408494527483855314145088375$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.40112868190846756728866065531$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9400617403938522$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.8953725084636996$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.2584481583354583883252841080$ |
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| Real period: | $\Omega$ | ≈ | $0.37949676281402848464294836988$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.8206160180606593238936501519 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.820616018 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.379497 \cdot 1.258448 \cdot 32}{2^2} \\ & \approx 3.820616018\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 65536 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{8}^{*}$ | additive | 1 | 6 | 18 | 0 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 | 8.24.0.69 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7280 = 2^{4} \cdot 5 \cdot 7 \cdot 13 \), index $192$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4175 & 166 \\ 474 & 4067 \end{array}\right),\left(\begin{array}{rr} 4096 & 3201 \\ 3185 & 4096 \end{array}\right),\left(\begin{array}{rr} 7269 & 6354 \\ 2530 & 3349 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 4 & 49 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 16 \\ 64 & 205 \end{array}\right),\left(\begin{array}{rr} 1457 & 16 \\ 5828 & 65 \end{array}\right),\left(\begin{array}{rr} 561 & 16 \\ 2244 & 65 \end{array}\right),\left(\begin{array}{rr} 7265 & 16 \\ 7264 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[7280])$ is a degree-$3246202552320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7280\Z)$.
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 | $2$ | \( 1 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 5824 = 2^{6} \cdot 7 \cdot 13 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 4160 = 2^{6} \cdot 5 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 2240 = 2^{6} \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 29120a
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 455b4, its twist by $8$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\zeta_{8})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.157351936.1 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.292464640000.11 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.702207600640000.40 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | ss | nonsplit | nonsplit | ss | nonsplit | ord | ss | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 1,1 | 1 | 1 | 3,1 | 1 | 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 | 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.