Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-1008161x-389747361\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-1008161xz^2-389747361z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-81661068x-283880842992\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-575, 408)$ | $3.6659548944357575011725540435$ | $\infty$ |
| $(-569, 0)$ | $0$ | $2$ |
| $(1159, 0)$ | $0$ | $2$ |
Integral points
\( \left(-591, 0\right) \), \((-575,\pm 408)\), \( \left(-569, 0\right) \), \( \left(1159, 0\right) \), \((5767,\pm 430848)\)
Invariants
| Conductor: | $N$ | = | \( 73920 \) | = | $2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $70815596544000000$ | = | $2^{20} \cdot 3^{6} \cdot 5^{6} \cdot 7^{2} \cdot 11^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{432288716775559561}{270140062500} \) | = | $2^{-2} \cdot 3^{-6} \cdot 5^{-6} \cdot 7^{-2} \cdot 11^{-2} \cdot 31^{3} \cdot 24391^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1751219345184066696843524444$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1354011636784887055585042622$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9765179895036278$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.735149103597227$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.6659548944357575011725540435$ |
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| Real period: | $\Omega$ | ≈ | $0.15067227147542292867603339137$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 192 $ = $ 2^{2}\cdot( 2 \cdot 3 )\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.6282930128529583033660864753 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.628293013 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.150672 \cdot 3.665955 \cdot 192}{4^2} \\ & \approx 6.628293013\end{aligned}$$
Modular invariants
Modular form 73920.2.a.eo
For more coefficients, see the Downloads section to the right.
| Modular degree: | 884736 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{10}^{*}$ | additive | -1 | 6 | 20 | 2 |
| $3$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $5$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $11$ | $2$ | $I_{2}$ | nonsplit 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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9240 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 2303 & 9228 \\ 4686 & 113 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 9224 & 9233 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 9229 & 12 \\ 9228 & 13 \end{array}\right),\left(\begin{array}{rr} 3081 & 4 \\ 7702 & 9 \end{array}\right),\left(\begin{array}{rr} 8407 & 6 \\ 3354 & 9235 \end{array}\right),\left(\begin{array}{rr} 3697 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4619 & 9228 \\ 9234 & 9167 \end{array}\right),\left(\begin{array}{rr} 5281 & 6 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[9240])$ is a degree-$2452488192000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9240\Z)$.
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 | $2$ | \( 1 \) |
| $3$ | split multiplicative | $4$ | \( 4928 = 2^{6} \cdot 7 \cdot 11 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 14784 = 2^{6} \cdot 3 \cdot 7 \cdot 11 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 10560 = 2^{6} \cdot 3 \cdot 5 \cdot 11 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 73920gg
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 2310l2, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{22})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-22}, \sqrt{-70})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{70})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $6$ | 6.2.1943822555136.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.1439868559360000.70 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.7965941760000.35 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.4857532416.2 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $18$ | 18.0.133393098492973037557429545975939072000000000000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | split | nonsplit | nonsplit | nonsplit | ord | ord | ord | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 2 | 1 | 1 | 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 |
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.