Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-4385x+30177\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-4385xz^2+30177z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-355212x+20933424\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-23, 344)$ | $0$ | $4$ |
Integral points
\((-23,\pm 344)\), \( \left(63, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 41280 \) | = | $2^{6} \cdot 3 \cdot 5 \cdot 43$ |
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| Discriminant: | $\Delta$ | = | $5041227202560$ | = | $2^{15} \cdot 3^{2} \cdot 5 \cdot 43^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{284630612552}{153846045} \) | = | $2^{3} \cdot 3^{-2} \cdot 5^{-1} \cdot 11^{3} \cdot 13^{3} \cdot 23^{3} \cdot 43^{-4}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1284094169203807957944963770$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.26197544122044915902295622518$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9326599658846131$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.4598423264153206$ | |||
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: | $\Omega$ | ≈ | $0.66985469116224086109910273440$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.3397093823244817221982054688 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.339709382 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.669855 \cdot 1.000000 \cdot 32}{4^2} \\ & \approx 1.339709382\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 73728 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 6 | 15 | 0 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $43$ | $4$ | $I_{4}$ | 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 | 4.12.0.7 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1720 = 2^{3} \cdot 5 \cdot 43 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1504 & 637 \\ 1071 & 1042 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1714 & 1715 \end{array}\right),\left(\begin{array}{rr} 348 & 1 \\ 711 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 1513 & 1508 \\ 1510 & 647 \end{array}\right),\left(\begin{array}{rr} 1121 & 8 \\ 1044 & 33 \end{array}\right),\left(\begin{array}{rr} 1713 & 8 \\ 1712 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[1720])$ is a degree-$51263815680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1720\Z)$.
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 | $4$ | \( 5 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 13760 = 2^{6} \cdot 5 \cdot 43 \) |
| $5$ | split multiplicative | $6$ | \( 8256 = 2^{6} \cdot 3 \cdot 43 \) |
| $43$ | split multiplicative | $44$ | \( 960 = 2^{6} \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 41280cs
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 20640c2, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{10}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.18933760.4 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.21233664000000.33 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.8962181693440000.74 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 43 |
|---|---|---|---|---|
| Reduction type | add | nonsplit | split | split |
| $\lambda$-invariant(s) | - | 0 | 1 | 1 |
| $\mu$-invariant(s) | - | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ 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 $0$.