Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-9441x-307359\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-9441xz^2-307359z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-764748x-226358928\)
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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 |
|---|---|---|
| $(-64, 175)$ | $1.7324902427453614640821405292$ | $\infty$ |
| $(-39, 0)$ | $0$ | $2$ |
| $(111, 0)$ | $0$ | $2$ |
Integral points
\( \left(-71, 0\right) \), \((-64,\pm 175)\), \((-45,\pm 156)\), \( \left(-39, 0\right) \), \( \left(111, 0\right) \), \((153,\pm 1344)\), \((761,\pm 20800)\)
Invariants
| Conductor: | $N$ | = | \( 87360 \) | = | $2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $12210831360000$ | = | $2^{18} \cdot 3^{2} \cdot 5^{4} \cdot 7^{2} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{355045312441}{46580625} \) | = | $3^{-2} \cdot 5^{-4} \cdot 7^{-2} \cdot 13^{-2} \cdot 73^{3} \cdot 97^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2392708888919493589780501796$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.19955011805203139485220199741$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8881027733274627$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.4340720759138823$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.7324902427453614640821405292$ |
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| Real period: | $\Omega$ | ≈ | $0.48852357705817440918303926018$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.3854493224173955155138239444 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.385449322 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.488524 \cdot 1.732490 \cdot 64}{4^2} \\ & \approx 3.385449322\end{aligned}$$
Modular invariants
Modular form 87360.2.a.c
For more coefficients, see the Downloads section to the right.
| Modular degree: | 196608 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{8}^{*}$ | additive | 1 | 6 | 18 | 0 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_{2}$ | 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 | 8.12.0.1 |
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
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 471 & 2 \\ 778 & 1091 \end{array}\right),\left(\begin{array}{rr} 591 & 2 \\ 334 & 1091 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1637 & 2182 \\ 0 & 2183 \end{array}\right),\left(\begin{array}{rr} 1819 & 2180 \\ 1454 & 2175 \end{array}\right),\left(\begin{array}{rr} 1091 & 0 \\ 0 & 2183 \end{array}\right),\left(\begin{array}{rr} 2181 & 4 \\ 2180 & 5 \end{array}\right),\left(\begin{array}{rr} 1095 & 2180 \\ 1096 & 2179 \end{array}\right)$.
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 | $2$ | \( 1 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 29120 = 2^{6} \cdot 5 \cdot 7 \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 17472 = 2^{6} \cdot 3 \cdot 7 \cdot 13 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 12480 = 2^{6} \cdot 3 \cdot 5 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 87360e
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1365f2, 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 |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-2}, \sqrt{91})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{6}, \sqrt{182})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-3})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | 16.0.132513778481397717569346994176.21 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 |
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 | nonsplit | nonsplit | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 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 |
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.