Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-454x-544\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-454xz^2-544z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-587763x-23605938\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{87}{4}, -\frac{91}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([174:-91:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(786, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 30 \) | = | $2 \cdot 3 \cdot 5$ |
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| Minimal Discriminant: | $\Delta$ | = | $5859375000$ | = | $2^{3} \cdot 3 \cdot 5^{12} $ |
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| j-invariant: | $j$ | = | \( \frac{10316097499609}{5859375000} \) | = | $2^{-3} \cdot 3^{-1} \cdot 5^{-12} \cdot 11^{3} \cdot 1979^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.56416598521635168028924232156$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.56416598521635168028924232156$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1359972360691755$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $8.810052252997776$ | |||
| 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: | $\Omega$ | ≈ | $1.1173160864138321498160760446$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.55865804320691607490803802232 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.558658043 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.117316 \cdot 1.000000 \cdot 2}{2^2} \\ & \approx 0.558658043\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 24 |
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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 semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
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 | 8.12.0.8 | $12$ |
| $3$ | 3B.1.2 | 3.8.0.2 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 1 & 12 \\ 12 & 25 \end{array}\right),\left(\begin{array}{rr} 96 & 113 \\ 71 & 114 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 97 & 24 \\ 96 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 6 & 49 \\ 35 & 56 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 14 & 11 \end{array}\right),\left(\begin{array}{rr} 94 & 3 \\ 15 & 34 \end{array}\right),\left(\begin{array}{rr} 97 & 24 \\ 84 & 49 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$92160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 3 \) |
| $3$ | split multiplicative | $4$ | \( 1 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 6 = 2 \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 30a
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.2.24.1-150.1-e5 |
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/4\Z\) | 2.2.8.1-450.1-a3 |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/4\Z\) | 2.2.12.1-150.1-a3 |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/6\Z\) | 2.0.3.1-300.1-a3 |
| $3$ | 3.1.243.1 | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{3})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-3})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-3})\) | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\zeta_{12})\) | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.177147.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.90699264.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.30233088.6 | \(\Z/12\Z\) | not in database |
| $6$ | 6.2.11337408.2 | \(\Z/12\Z\) | not in database |
| $8$ | 8.0.3057647616.9 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.8.23592960000.1 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.1866240000.6 | \(\Z/8\Z\) | not in database |
| $8$ | \(\Q(\zeta_{24})\) | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | 12.0.8226356490141696.17 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.8226356490141696.32 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $12$ | 12.0.128536820158464.4 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $12$ | 12.4.526486815369068544.25 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | 16.0.149587343098087735296.14 | \(\Z/4\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.0.891610044825600000000.6 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | 16.0.3482851737600000000.5 | \(\Z/24\Z\) | not in database |
| $18$ | 18.0.617673396283947000000000000.3 | \(\Z/18\Z\) | 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 |
|---|---|---|---|
| Reduction type | nonsplit | split | nonsplit |
| $\lambda$-invariant(s) | 0 | 1 | 0 |
| $\mu$-invariant(s) | 0 | 1 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.