Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+x^2-30004408x-63144368812\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+x^2z-30004408xz^2-63144368812z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-2430357075x-46024953792750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-3178, 11088)$ | $0.46510810185752777260361899556$ | $\infty$ |
| $(-3277, 0)$ | $0$ | $2$ |
Integral points
\( \left(-3277, 0\right) \), \((-3178,\pm 11088)\), \((-3052,\pm 3150)\), \((6524,\pm 137214)\), \((10198,\pm 831600)\), \((18998,\pm 2494800)\), \((113687,\pm 38287242)\)
Invariants
| Conductor: | $N$ | = | \( 92400 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11$ |
|
| Discriminant: | $\Delta$ | = | $6832189170132480000000$ | = | $2^{15} \cdot 3^{12} \cdot 5^{7} \cdot 7^{3} \cdot 11^{4} $ |
|
| j-invariant: | $j$ | = | \( \frac{46676570542430835889}{106752955783320} \) | = | $2^{-3} \cdot 3^{-12} \cdot 5^{-1} \cdot 7^{-3} \cdot 11^{-4} \cdot 47^{3} \cdot 76607^{3}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0709816236443704159449428266$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.5731154868673749192273310385$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9957188233471589$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.533043477890861$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.46510810185752777260361899556$ |
|
| Real period: | $\Omega$ | ≈ | $0.064515337012777336789471789430$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1152 $ = $ 2^{2}\cdot( 2^{2} \cdot 3 )\cdot2\cdot3\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $8.6419025103489330520302995088 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 8.641902510 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.064515 \cdot 0.465108 \cdot 1152}{2^2} \\ & \approx 8.641902510\end{aligned}$$
Modular invariants
Modular form 92400.2.a.ho
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7962624 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{7}^{*}$ | additive | -1 | 4 | 15 | 3 |
| $3$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $5$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $11$ | $4$ | $I_{4}$ | split 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 | 8.12.0.7 |
| $3$ | 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
$\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 4621 & 24 \\ 3092 & 1829 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 1926 & 409 \\ 4235 & 2696 \end{array}\right),\left(\begin{array}{rr} 9217 & 24 \\ 9216 & 25 \end{array}\right),\left(\begin{array}{rr} 2521 & 24 \\ 2532 & 289 \end{array}\right),\left(\begin{array}{rr} 5528 & 9237 \\ 6003 & 86 \end{array}\right),\left(\begin{array}{rr} 2306 & 9237 \\ 2265 & 9206 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 7934 & 11 \end{array}\right),\left(\begin{array}{rr} 1336 & 21 \\ 1035 & 8866 \end{array}\right)$.
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 | $4$ | \( 175 = 5^{2} \cdot 7 \) |
| $3$ | split multiplicative | $4$ | \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \) |
| $5$ | additive | $18$ | \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \) |
| $7$ | split multiplicative | $8$ | \( 13200 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 92400.ho
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 2310.u4, its twist by $-20$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{70}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-35}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-35})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{-14})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{7})\) | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-5})\) | \(\Z/12\Z\) | not in database |
| $6$ | 6.2.79061400000.2 | \(\Z/6\Z\) | not in database |
| $8$ | 8.4.30840979456000000.14 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.98344960000.4 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | 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 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.0.12510813651626104030288607026883973120000000000000.2 | \(\Z/18\Z\) | 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 | add | split | split | ord | ord | ord | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 4 | - | 2 | 2 | 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 |
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.