Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-4524x-15210\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-4524xz^2-15210z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-366471x-12187476\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(131, 1274)$ | $3.1379076158195511766732782874$ | $\infty$ |
| $(-65, 0)$ | $0$ | $2$ |
Integral points
\( \left(-65, 0\right) \), \((131,\pm 1274)\)
Invariants
| Conductor: | $N$ | = | \( 122304 \) | = | $2^{6} \cdot 3 \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $5806379097792$ | = | $2^{6} \cdot 3^{3} \cdot 7^{6} \cdot 13^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{1360251712}{771147} \) | = | $2^{6} \cdot 3^{-3} \cdot 13^{-4} \cdot 277^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1390978265970653682396432598$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.18043083821056393902164917265$ |
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| $abc$ quality: | $Q$ | ≈ | $1.069422794221752$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.147041619150247$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.1379076158195511766732782874$ |
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| Real period: | $\Omega$ | ≈ | $0.62820340713664622030069311093$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot1\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.9424885110757447235030268385 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.942488511 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.628203 \cdot 3.137908 \cdot 8}{2^2} \\ & \approx 3.942488511\end{aligned}$$
Modular invariants
Modular form 122304.2.a.x
For more coefficients, see the Downloads section to the right.
| Modular degree: | 294912 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | 1 | 6 | 6 | 0 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $4$ | $I_{4}$ | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2184 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 2177 & 8 \\ 2176 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 120 & 1841 \\ 889 & 876 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 2017 & 1568 \\ 1204 & 1905 \end{array}\right),\left(\begin{array}{rr} 1772 & 1561 \\ 1687 & 1254 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 2178 & 2179 \end{array}\right),\left(\begin{array}{rr} 1357 & 1050 \\ 1750 & 349 \end{array}\right),\left(\begin{array}{rr} 935 & 0 \\ 0 & 2183 \end{array}\right)$.
The torsion field $K:=\Q(E[2184])$ is a degree-$81155063808$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2184\Z)$.
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 | $2$ | \( 147 = 3 \cdot 7^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 40768 = 2^{6} \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $26$ | \( 2496 = 2^{6} \cdot 3 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 122304cd
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1248h3, its twist by $-56$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-14}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-42}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{-14})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.458838245376.19 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.2588618098999296.19 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | nonsplit | ord | add | ord | split | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 1 | 1 | - | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3,1 |
| $\mu$-invariant(s) | - | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.