Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-10283859x-12690955580\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-10283859xz^2-12690955580z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-164541747x-812385698866\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{7405}{4}, \frac{7405}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-14810:7405:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-7406, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 23805 \) | = | $3^{2} \cdot 5 \cdot 23^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $43706856047805$ | = | $3^{10} \cdot 5 \cdot 23^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{1114544804970241}{405} \) | = | $3^{-4} \cdot 5^{-1} \cdot 103681^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4079228607757312130746346587$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.29086960847710152197363562433$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0735374703334082$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.958914628458147$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
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: | $\Omega$ | ≈ | $0.084306396042867003897945947679$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.3489023366858720623671351629 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $16$ = $4^2$ (exact) |
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BSD formula
$$\begin{aligned} 1.348902337 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{16 \cdot 0.084306 \cdot 1.000000 \cdot 4}{2^2} \\ & \approx 1.348902337\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 405504 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $23$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 16.48.0.121 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 11040 = 2^{5} \cdot 3 \cdot 5 \cdot 23 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 23 & 18 \\ 8478 & 9035 \end{array}\right),\left(\begin{array}{rr} 137 & 10534 \\ 138 & 5749 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 6832 & 7705 \\ 23 & 1266 \end{array}\right),\left(\begin{array}{rr} 4319 & 0 \\ 0 & 11039 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 1381 & 1472 \\ 7636 & 1473 \end{array}\right),\left(\begin{array}{rr} 11009 & 32 \\ 11008 & 33 \end{array}\right),\left(\begin{array}{rr} 4831 & 1472 \\ 7590 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[11040])$ is a degree-$3151641968640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/11040\Z)$.
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 |
|---|---|---|---|
| $3$ | additive | $8$ | \( 2645 = 5 \cdot 23^{2} \) |
| $5$ | split multiplicative | $6$ | \( 4761 = 3^{2} \cdot 23^{2} \) |
| $23$ | additive | $266$ | \( 45 = 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 23805t
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 15a5, its twist by $69$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-345}) \) | \(\Z/4\Z\) | 2.0.1380.1-15.1-f8 |
| $2$ | \(\Q(\sqrt{-69}) \) | \(\Z/4\Z\) | 2.0.276.1-75.2-a8 |
| $4$ | \(\Q(\sqrt{5}, \sqrt{-69})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-69})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{10}, \sqrt{-69})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.4.90668484000000.15 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.23211131904000000.42 | \(\Z/8\Z\) | not in database |
| $8$ | \(\Q(\sqrt{2}, \sqrt{5}, \sqrt{-69})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | 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 | 23 |
|---|---|---|---|---|
| Reduction type | ord | add | split | add |
| $\lambda$-invariant(s) | 5 | - | 1 | - |
| $\mu$-invariant(s) | 3 | - | 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 $0$.