Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+93x-424\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+93xz^2-424z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+1485x-25650\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(8, 24\right) \) | $1.7202730826931517149229941709$ | $\infty$ |
| \( \left(4, -2\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([8:24:1]\) | $1.7202730826931517149229941709$ | $\infty$ |
| \([4:-2:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(31, 224\right) \) | $1.7202730826931517149229941709$ | $\infty$ |
| \( \left(15, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(4, -2\right) \), \( \left(8, 24\right) \), \( \left(8, -32\right) \)
\([4:-2:1]\), \([8:24:1]\), \([8:-32:1]\)
\( \left(15, 0\right) \), \((31,\pm 224)\)
Invariants
| Conductor: | $N$ | = | \( 1575 \) | = | $3^{2} \cdot 5^{2} \cdot 7$ |
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| Minimal Discriminant: | $\Delta$ | = | $-120558375$ | = | $-1 \cdot 3^{9} \cdot 5^{3} \cdot 7^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{35937}{49} \) | = | $3^{3} \cdot 7^{-2} \cdot 11^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.23636515171559866908877000377$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.98995354289400869310785375723$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8394167977331788$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.465082927342656$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.7202730826931517149229941709$ |
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| Real period: | $\Omega$ | ≈ | $0.99163738727769693640523243831$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.4117742102519729084501603222 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.411774210 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.991637 \cdot 1.720273 \cdot 8}{2^2} \\ & \approx 3.411774210\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 384 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $5$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $7$ | $2$ | $I_{2}$ | split 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.12.0.31 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \), index $96$, genus $3$, and generators
$\left(\begin{array}{rr} 3 & 28 \\ 1604 & 971 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 1616 & 1587 \end{array}\right),\left(\begin{array}{rr} 1136 & 5 \\ 51 & 16 \end{array}\right),\left(\begin{array}{rr} 1205 & 26 \\ 1334 & 1129 \end{array}\right),\left(\begin{array}{rr} 13 & 8 \\ 8 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1471 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1665 & 16 \\ 1664 & 17 \end{array}\right),\left(\begin{array}{rr} 421 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1016 & 5 \\ 387 & 32 \end{array}\right)$.
The torsion field $K:=\Q(E[1680])$ is a degree-$11890851840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1680\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 15 = 3 \cdot 5 \) |
| $3$ | additive | $2$ | \( 175 = 5^{2} \cdot 7 \) |
| $5$ | additive | $10$ | \( 63 = 3^{2} \cdot 7 \) |
| $7$ | split multiplicative | $8$ | \( 225 = 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 1575c
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1575d1, its twist by $-3$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15 +4 \sqrt{15}})\) | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.437582250000.4 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.2916000000.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.82046671875.3 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | ord | add | add | split | ss | ord | ord | ss | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | - | - | 2 | 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 | 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.